In this section, we describe functions related to general number fields. Functions related to quadratic number fields are found in Section se:arithmetic (Arithmetic functions).

Let K = ℚ[X] / (T) a number field, ℤ_{K} its ring of integers, T ∈ ℤ[X]
is monic. Three basic number field structures can be attached to K in
GP:

***** *nf* denotes a number field, i.e. a data structure output by
`nfinit`

. This contains the basic arithmetic data attached to the
number field: signature, maximal order (given by a basis `nf.zk`

),
discriminant, defining polynomial T, etc.

***** *bnf* denotes a "Buchmann's number field", i.e. a
data structure output by `bnfinit`

. This contains
*nf* and the deeper invariants of the field: units U(K), class group
Cl(K), as well as technical data required to solve the two attached
discrete logarithm problems.

***** *bnr* denotes a "ray number field", i.e. a data structure
output by `bnrinit`

, corresponding to the ray class group structure of
the field, for some modulus f. It contains a *bnf*, the modulus
f, the ray class group Cl_{f}(K) and data attached to
the discrete logarithm problem therein.

An *algebraic number* belonging to K = ℚ[X]/(T) is given as

***** a `t_INT`

, `t_FRAC`

or `t_POL`

(implicitly modulo T), or

***** a `t_POLMOD`

(modulo T), or

***** a `t_COL`

`v`

of dimension N = [K:ℚ], representing
the element in terms of the computed integral basis, as
`sum(i = 1, N, v[i] * nf.zk[i])`

. Note that a `t_VEC`

will not be recognized.

An *ideal* is given in any of the following ways:

***** an algebraic number in one of the above forms, defining a principal ideal.

***** a prime ideal, i.e. a 5-component vector in the format output by
`idealprimedec`

or `idealfactor`

.

***** a `t_MAT`

, square and in Hermite Normal Form (or at least
upper triangular with nonnegative coefficients), whose columns represent a
ℤ-basis of the ideal.

One may use `idealhnf`

to convert any ideal to the last (preferred) format.

***** an *extended ideal* is a 2-component
vector [I, t], where I is an ideal as above and t is an algebraic
number, representing the ideal (t)I. This is useful whenever `idealred`

is involved, implicitly working in the ideal class group, while keeping track
of principal ideals. The following multiplicative ideal operations
update the principal part: `idealmul`

, `idealinv`

,
`idealsqr`

, `idealpow`

and `idealred`

; e.g. using `idealmul`

on [I,t], [J,u], we obtain [IJ, tu]. In all other
functions, the extended part is silently discarded, e.g. using
`idealadd`

with the above input produces I+J.

The "principal part" t in an extended ideal may be
represented in any of the above forms, and *also* as a factorization
matrix (in terms of number field elements, not ideals!), possibly the empty
factorization matrix `factor(1)`

representing 1; the empty matrix
`[;]`

is also accepted as a synonym for 1. When t is such a
factorization matrix, elements stay in
factored form, or *famat* for *fa*ctorization *mat*rix, which
is a convenient way to avoid coefficient explosion. To recover the
conventional expanded form, try `nffactorback`

; but many functions
already accept *famat*s as input, for instance `ideallog`

, so
expanding huge elements should never be necessary.

A finite abelian group G in user-readable format is given by its Smith
Normal Form as a pair [h,d] or triple [h,d,g].
Here h is the cardinality of G, (d_{i}) is the vector of elementary
divisors, and (g_{i}) is a vector of generators. In short,
G = ⨁ _{i ≤ n} (ℤ/d_{i}ℤ) g_{i}, with d_{n} | ... | d_{2} | d_{1}
and ∏ d_{i} = h. This information can also be retrieved as
G.`no`

, G.`cyc`

and G.`gen`

.

***** a *character* on the abelian group
⨁ (ℤ/d_{j}ℤ) g_{j}
is given by a row vector χ = [a_{1},...,a_{n}] such that
χ(∏ g_{j}^{nj}) = exp(2π i∑ a_{j} n_{j} / d_{j}).

***** given such a structure, a *subgroup* H is input as a square
matrix in HNF, whose columns express generators of H on the given generators
g_{i}. Note that the determinant of that matrix is equal to the index (G:H).

We now have a look at data structures attached to relative extensions
of number fields L/K, and to projective ℤ_{K}-modules. When defining a
relative extension L/K, the *nf* attached to the base field K
must be defined by a variable having a lower priority (see
Section se:priority) than the variable defining the extension. For example,
you may use the variable name y to define the base field K, and x to
define the relative extension L/K.

**Basic definitions.**

***** *rnf* denotes a relative number field, i.e. a data structure
output by `rnfinit`

, attached to the extension L/K. The *nf*
attached to be base field K is `rnf.nf`

.

***** A *relative matrix* is an m x n matrix whose entries are
elements of K, in any form. Its m columns A_{j} represent elements
in K^n.

***** An *ideal list* is a row vector of fractional ideals of the number
field *nf*.

***** A *pseudo-matrix* is a 2-component row vector (A,I) where A
is a relative m x n matrix and I an ideal list of length n. If I =
{𝔞_{1},..., 𝔞_{n}} and the columns of A are (A_{1},...,
A_{n}), this data defines the torsion-free (projective) ℤ_{K}-module
𝔞_{1} A_{1}⨁ 𝔞_{n} A_{n}.

***** An *integral pseudo-matrix* is a 3-component row vector (A,I,J)
where A = (a_{i,j}) is an m x n relative matrix and I =
(𝔟_{1},..., 𝔟_{m}), J = (𝔞_{1},..., 𝔞_{n}) are ideal
lists, such that a_{i,j} ∈ 𝔟_{i} 𝔞_{j}^{-1} for all i,j. This
data defines two abstract projective ℤ_{K}-modules
N = 𝔞_{1}ω_{1}⨁ ...⨁ 𝔞_{n}ω_{n} in K^n,
P = 𝔟_{1}η_{1}⨁ ...⨁ 𝔟_{m}η_{m} in K^m, and a
ℤ_{K}-linear map f:N → P given by
f(∑ α_{j}ω_{j}) = ∑_{i} (a_{i,j}α_{j}) η_{i}.
This data defines the ℤ_{K}-module M = P/f(N).

***** Any *projective* ℤ_{K}-moduleprojective module M
of finite type in K^m can be given by a pseudo matrix (A,I).

***** An arbitrary ℤ_{K} module of finite type in K^m, with nontrivial
torsion, is given by an integral pseudo-matrix (A,I,J)

**Algebraic numbers in relative extension.**

We are given a number field K = `nfinit`

(T), attached to K = ℚ[Y]/(T),
T ∈ ℚ[Y], and a relative extension L = `rnfinit`

(K, P), attached
to L = K[X]/(P), P ∈ K[X].
In all contexts (except `rnfeltabstorel`

, see below), an
*algebraic number* is given as

***** a `t_INT`

, `t_FRAC`

or `t_POL`

in ℚ[Y] (implicitly modulo T)
or a `t_POL`

in K[X] (implicitly modulo P),

***** a `t_POLMOD`

(modulo T or P), or

***** a `t_COL`

`v`

of dimension m = [K:ℚ], representing
the element in terms of the integral basis `K.zk`

;

***** if an absolute `nf`

structure `Labs`

was attached to L, via
`Labs = nfinit`

(L), then we can also use a `t_COL`

`v`

of
dimension [L:ℚ], representing the element in terms of the computed integral
basis `Labs.zk`

. Be careful that in the degenerate case
L = K, then the previous interpretation (with respect to `K.zk`

)
takes precedence. This is no concern when K = ℚ or if P = X - Y
(because in that case the primitive
polynomial `Labs.pol`

defining L of ℚ is `nf.pol`

and the
computation of `nf.zk`

is deterministic); but in other cases, the
integer bases attached to K and `Labs`

may differ.

**Special case: rnfabstorel.** This function assumes
that elements are given in absolute representation (with respect to

`Labs.zk`

or modulo `Labs.pol`

and converts them to relative
representation modulo `L.pol`

. In that function (only), a `t_POL`

in
X is implicitly understood modulo `Labs.pol`

and a `t_COL`

of length [L:ℚ] refers to the integral basis `Labs.zk`

in all cases,
including L = K.
**Pseudo-bases, determinant.**

***** The pair (A,I) is a *pseudo-basis* of the module it
generates if the 𝔞_{j} are nonzero, and the A_{j} are K-linearly
independent. We call n the *size* of the pseudo-basis. If A is a
relative matrix, the latter condition means it is square with nonzero
determinant; we say that it is in Hermite Normal
Form (HNF) if it is upper triangular and all the
elements of the diagonal are equal to 1.

***** For instance, the relative integer basis `rnf.zk`

is a pseudo-basis
(A,I) of ℤ_{L}, where A = `rnf.zk[1]`

is a vector of elements of L,
which are K-linearly independent. Most *rnf* routines return and handle
ℤ_{K}-modules contained in L (e.g. ℤ_{L}-ideals) via a pseudo-basis
(A',I'), where A' is a relative matrix representing a vector of elements of
L in terms of the fixed basis `rnf.zk[1]`

***** The *determinant* of a pseudo-basis (A,I) is the ideal
equal to the product of the determinant of A by all the ideals of I. The
determinant of a pseudo-matrix is the determinant of any pseudo-basis of the
module it generates.

A *modulus*, in the sense of class field theory, is a divisor supported
on the real and finite places of K. In PARI terms, this means either an
ordinary ideal I as above (no Archimedean component), or a pair [I,a],
where a is a vector with r_{1} {0,1}-components, corresponding to the
infinite part of the divisor. More precisely, the i-th component of a
corresponds to the real embedding attached to the i-th real root of
`K.roots`

. (That ordering is not canonical, but well defined once a
defining polynomial for K is chosen.) For instance, `[1, [1,1]]`

is a
modulus for a real quadratic field, allowing ramification at any of the two
places at infinity, and nowhere else.

A *bid* or "big ideal" is a structure output by `idealstar`

needed to compute in (ℤ_{K}/I)^{*}, where I is a modulus in the above sense.
It is a finite abelian group as described above, supplemented by
technical data needed to solve discrete log problems.

Finally we explain how to input ray number fields (or *bnr*), using class
field theory. These are defined by a triple A, B, C, where the
defining set [A,B,C] can have any of the following forms:
[*bnr*],
[*bnr*,*subgroup*],
[*bnr*,*character*],
[*bnf*,*mod*],
[*bnf*,*mod*,*subgroup*]. The last two forms are kept for
backward compatibility, but no longer serve any real purpose (see example
below); no newly written function will accept them.

***** *bnf* is as output by `bnfinit`

, where units are mandatory
unless the modulus is trivial; *bnr* is as output by `bnrinit`

. This
is the ground field K.

***** *mod* is a modulus 𝔣, as described above.

***** *subgroup* a subgroup of the ray class group modulo 𝔣 of
K. As described above, this is input as a square matrix expressing
generators of a subgroup of the ray class group

on the
given generators. We also allow a *bnr*.clgp`t_INT`

n for n.Cl_{f}.

***** *character* is a character χ of the ray class group modulo
𝔣, representing the subgroup Ker χ.

The corresponding *bnr* is the subfield of the ray class field of K
modulo 𝔣, fixed by the given subgroup.

? K = bnfinit(y^2+1); ? bnr = bnrinit(K, 13) ? %.clgp %3 = [36, [12, 3]] ? bnrdisc(bnr); \\ discriminant of the full ray class field ? bnrdisc(bnr, [3,1;0,1]); \\ discriminant of cyclic cubic extension of K ? bnrconductor(bnr, [3,1]); \\ conductor of chi: g1->zeta_12^3, g2->zeta_{3}

We could have written directly

? bnrdisc(K, 13); ? bnrdisc(K, 13, [3,1;0,1]);

avoiding one `bnrinit`

, but this would actually be slower since the
`bnrinit`

is called internally anyway. And now twice!

All the functions which are specific to relative extensions, number fields,
Buchmann's number fields, Buchmann's number rays, share the prefix `rnf`

,
`nf`

, `bnf`

, `bnr`

respectively. They take as first argument a
number field of that precise type, respectively output by `rnfinit`

,
`nfinit`

, `bnfinit`

, and `bnrinit`

.

However, and even though it may not be specified in the descriptions of the
functions below, it is permissible, if the function expects a *nf*, to
use a *bnf* instead, which contains much more information. On the other
hand, if the function requires a `bnf`

, it will *not* launch
`bnfinit`

for you, which is a costly operation. Instead, it will give you
a specific error message. In short, the types
`nf`

≤ `bnf`

≤ `bnr`

are ordered, each function requires a minimal type to work properly, but you
may always substitute a larger type.

The data types corresponding to the structures described above are rather complicated. Thus, as we already have seen it with elliptic curves, GP provides "member functions" to retrieve data from these structures (once they have been initialized of course). The relevant types of number fields are indicated between parentheses:

`bid`

(*bnr* ) : bid ideal structure.

`bnf`

(*bnr*, *bnf* ) : Buchmann's number field.

`clgp`

(*bnr*, *bnf* ) : classgroup. This one admits the
following three subclasses:

`cyc`

: cyclic decomposition
(SNF).

`gen`

:
generators.

`no`

: number of elements.

`diff`

(*bnr*, *bnf*, *nf* ) : the different ideal.

`codiff`

(*bnr*, *bnf*, *nf* ) : the codifferent
(inverse of the different in the ideal group).

`disc`

(*bnr*, *bnf*, *nf* ) : discriminant.

`fu`

( *bnf* ) : fundamental units.

`index`

(*bnr*, *bnf*, *nf* ) :
index of the power order in the ring of integers.

`mod`

(*bnr* ) : modulus.

`nf`

(*bnr*, *bnf*, *nf* ) : number field.

`pol`

(*bnr*, *bnf*, *nf* ) : defining polynomial.

`r1`

(*bnr*, *bnf*, *nf* ) : the number
of real embeddings.

`r2`

(*bnr*, *bnf*, *nf* ) : the number
of pairs of complex embeddings.

`reg`

( *bnf* ) : regulator.

`roots`

(*bnr*, *bnf*, *nf* ) : roots of the
polynomial generating the field.

`sign`

(*bnr*, *bnf*, *nf* ) : signature [r1,r2].

`t2`

(*bnr*, *bnf*, *nf* ) : the T_{2} matrix (see
`nfinit`

).

`tu`

( *bnf* ) : a generator for the torsion
units.

`zk`

(*bnr*, *bnf*, *nf* ) : integral basis, i.e. a
ℤ-basis of the maximal order.

`zkst`

(*bnr* ) : structure of (ℤ_{K}/m)^{*}.

The member functions `.codiff`

, `.t2`

and `.zk`

perform a
computation and are relatively expensive in large degree: move them out of
tight loops and store them in variables.

For instance, assume that *bnf* = `bnfinit`

(*pol*), for some
polynomial. Then

retrieves the class group, and
*bnf*.clgp

the class number. If we had set *bnf*.clgp.no*bnf* =
`nfinit`

(*pol*), both would have output an error message. All these
functions are completely recursive, thus for instance

will yield the maximal order of *bnr*.bnf.nf.zk*bnr*, which
you could get directly with a simple

.*bnr*.zk

Some of the functions starting with `bnf`

are implementations of the
sub-exponential algorithms for finding class and unit groups under GRH,
due to Hafner-McCurley, Buchmann and Cohen-Diaz-Olivier. The general
call to the functions concerning class groups of general number fields
(i.e. excluding `quadclassunit`

) involves a polynomial P and a
technical vector
*tech* = [c_{1}, c_{2}, *nrpid* ],
where the parameters are to be understood as follows:

P is the defining polynomial for the number field, which must be in
ℤ[X], irreducible and monic. In fact, if you supply a nonmonic polynomial
at this point, `gp`

issues a warning, then *transforms your
polynomial* so that it becomes monic. The `nfinit`

routine
will return a different result in this case: instead of `res`

, you get a
vector `[res,Mod(a,Q)]`

, where `Mod(a,Q) = Mod(X,P)`

gives the change
of variables. In all other routines, the variable change is simply lost.

The *tech* interface is obsolete and you should not tamper with
these parameters. Indeed, from version 2.4.0 on,

***** the results are always rigorous under GRH (before that version,
they relied on a heuristic strengthening, hence the need for overrides).

***** the influence of these parameters on execution time and stack size is
marginal. They *can* be useful to fine-tune and experiment with the
`bnfinit`

code, but you will be better off modifying all tuning
parameters in the C code (there are many more than just those three).
We nevertheless describe it for completeness.

The numbers c_{1} ≤ c_{2} are nonnegative real numbers. By default they are
chosen so that the result is correct under GRH. For i = 1,2, let
B_{i} = c_{i}(log |d_{K}|)^2, and denote by S(B) the set of maximal ideals of
K whose norm is less than B. We want S(B_{1}) to generate Cl(K) and hope
that S(B_{2}) can be *proven* to generate Cl(K).

More precisely, S(B_{1}) is a factorbase used to compute a tentative
Cl(K) by generators and relations. We then check explicitly, using
essentially `bnfisprincipal`

, that the elements of S(B_{2}) belong to the
span of S(B_{1}). Under the assumption that S(B_{2}) generates Cl(K), we
are done. User-supplied c_{i} are only used to compute initial guesses for
the bounds B_{i}, and the algorithm increases them until one can *prove*
under GRH that S(B_{2}) generates Cl(K). A uniform result of Bach says
that c_{2} = 12 is always suitable, but this bound is very pessimistic and a
direct algorithm due to Belabas-Diaz-Friedman is used to check the condition,
assuming GRH. The default values are c_{1} = c_{2} = 0. When c_{1} is equal to
0 the algorithm takes it equal to c_{2}.

*nrpid* is the maximal number of small norm relations attached to each
ideal in the factor base. Set it to 0 to disable the search for small norm
relations. Otherwise, reasonable values are between 4 and 20. The default is
4.

**Warning.** Make sure you understand the above! By default, most of
the `bnf`

routines depend on the correctness of the GRH. In particular,
any of the class number, class group structure, class group generators,
regulator and fundamental units may be wrong, independently of each other.
Any result computed from such a `bnf`

may be wrong. The only guarantee is
that the units given generate a subgroup of finite index in the full unit
group. You must use `bnfcertify`

to certify the computations
unconditionally.

**Remarks.**

You do not need to supply the technical parameters (under the library you
still need to send at least an empty vector, coded as `NULL`

). However,
should you choose to set some of them, they *must* be given in the
requested order. For example, if you want to specify a given value of
*nrpid*, you must give some values as well for c_{1} and c_{2}, and provide
a vector [c_{1},c_{2},*nrpid*].

Note also that you can use an *nf* instead of P, which avoids
recomputing the integral basis and analogous quantities.

Hecke Grossencharacters are continuous characters of the id\`ele class group;
they generalize classical Hecke characters on ray class groups obtained through
the *bnr* structure.

Let K be a number field, \A^ x its group of id\`eles. Every Grossencharacter

χ : \A^ x /K^ x → ℂ^ x

can be uniquely written χ = χ_{0} |.|^s for some s ∈ ℂ
and some character χ_{0} of the compact
group \A^ x /(K^ x.ℝ_{ > 0}),
where |a |= ∏_{v} |a_{v}|_{v} is the id\`ele norm.

Let 𝔪 be a modulus (an integral ideal and a finite set of real places). Let U(𝔪) be the subgroup of id\`eles congruent to 1 modulo 𝔪 (units outside 𝔪, positive at real places in 𝔪). The Hecke Grossencharacters defined modulo 𝔪 are the characters of the id\`ele class group

C_{K}(𝔪) = \A^ x /(K^ x.U(𝔪)),

that is, combinations of an archimedean
character χ_ oo on the connected component K_ oo ^{ x o }
and a ray class group character χ_{f} satisfying a compatibility condition
χ_ oo (a)χ_{f}(a) = 1 for all units a congruent to 1 modulo 𝔪.

gchar
***** *gc* denotes a structure allowing to compute with Hecke
Grossencharacters.

***** `gcharinit(`

initializes the structure *bnf*,*mod*)*gc*.
The underlying number field and modulus can be accessed using
*gc*`.bnf`

and *gc*`.mod`

.

***** *gc*`.cyc`

describes the finite abelian group structure of
*gc*, the torsion part corresponding to finite order ray class
characters, the exact zeros corresponding to a lattice of infinite order
Grossencharacters, and the approximate zero being a placeholder for the
complex powers of the id\`ele norm.

***** A Hecke character of modulus 𝔪 is described as a `t_COL`

of
coordinates corresponding to *gc*`.cyc`

: all the coordinates are
integers except the last one, which can be an arbitrary complex number, or
omitted instead of 0.

***** Hecke Grossencharacters have L-functions and can be given to all
`lfun`

functions as a 2 components vector `[`

, see
also Section se:lfungchar.*gc*,*chi*]

*bnf* being as output by
`bnfinit`

, checks whether the result is correct, i.e. whether it is
possible to remove the assumption of the Generalized Riemann
Hypothesis. It is correct if and only if the answer is 1. If it is
incorrect, the program may output some error message, or loop indefinitely.
You can check its progress by increasing the debug level. The *bnf*
structure must contain the fundamental units:

? K = bnfinit(x^3+2^2^3+1); bnfcertify(K) *** at top-level: K=bnfinit(x^3+2^2^3+1);bnfcertify(K) *** ^ — — — — - *** bnfcertify: precision too low in makeunits [use bnfinit(,1)]. ? K = bnfinit(x^3+2^2^3+1, 1); \\ include units ? bnfcertify(K) %3 = 1

If flag is present, only certify that the class group is a quotient of the one computed in bnf (much simpler in general); likewise, the computed units may form a subgroup of the full unit group. In this variant, the units are no longer needed:

? K = bnfinit(x^3+2^2^3+1); bnfcertify(K, 1) %4 = 1

The library syntax is `long `

.
Also available is **bnfcertify0**(GEN bnf, long flag)`GEN `

(**bnfcertify**(GEN bnf)*flag* = 0).

If m is a module as output in the
first component of an extension given by `bnrdisclist`

, outputs the
true module.

? K = bnfinit(x^2+23); L = bnrdisclist(K, 10); s = L[2] %1 = [[[Vecsmall([8]), Vecsmall([1])], [[0, 0, 0]]], [[Vecsmall([9]), Vecsmall([1])], [[0, 0, 0]]]] ? bnfdecodemodule(K, s[1][1]) %2 = [2 0] [0 1] ? bnfdecodemodule(K,s[2][1]) %3 = [2 1] [0 1]

The library syntax is `GEN `

.**decodemodule**(GEN nf, GEN m)

Initializes a
`bnf`

structure. Used in programs such as `bnfisprincipal`

,
`bnfisunit`

or `bnfnarrow`

. By default, the results are conditional
on the GRH, see se:GRHbnf. The result is a
10-component vector *bnf*.

This implements Buchmann's sub-exponential algorithm for computing the
class group, the regulator and a system of fundamental units of the
general algebraic number field K defined by the irreducible polynomial P
with integer coefficients. The meaning of *flag* is as follows:

***** *flag* = 0 (default). This is the historical behavior, kept for
compatibility reasons and speed. It has severe drawbacks but is likely to be
a little faster than the alternative, twice faster say, so only use it if
speed is paramount, you obtain a useful speed gain for the fields
under consideration, and you are only interested in the field invariants
such as the classgroup structure or its regulator. The computations involve
exact algebraic numbers which are replaced by floating point embeddings for
the sake of speed. If the precision is insufficient, `gp`

may not be able
to compute fundamental units, nor to solve some discrete logarithm problems.
It *may* be possible to increase the precision of the `bnf`

structure using `nfnewprec`

but this may fail, in particular when
fundamental units are large. In short, the resulting `bnf`

structure is correct and contains useful information but later
function calls to `bnfisprincpal`

or `bnrclassfield`

may fail.

When *flag* = 1, we keep an exact algebraic version of all floating point data
and this allows to guarantee that functions using the structure will always
succeed, as well as to compute the fundamental units exactly. The units are
computed in compact form, as a product of small S-units, possibly with
huge exponents. This flag also allows `bnfisprincipal`

to compute
generators of principal ideals in factored form as well. Be warned that
expanding such products explicitly can take a very long time, but they can
easily be mapped to floating point or ℓ-adic embeddings of bounded
accuracy, or to K^{*}/(K^{*})^ℓ, and this is enough for applications. In
short, this flag should be used by default, unless you have a very good
reason for it, for instance building massive tables of class numbers, and
you do not care about units or the effect large units would have on your
computation.

*tech* is a technical vector (empty by default, see se:GRHbnf).
Careful use of this parameter may speed up your computations,
but it is mostly obsolete and you should leave it alone.

The components of a *bnf* are technical.
In fact: *never access a component directly, always use
a proper member function.* However, for the sake of completeness and internal
documentation, their description is as follows. We use the notations
explained in the book by H. Cohen, *A Course in Computational Algebraic
Number Theory*, Graduate Texts in Maths **138**, Springer-Verlag, 1993,
Section 6.5, and subsection 6.5.5 in particular.

*bnf*[1] contains the matrix W, i.e. the matrix in Hermite normal
form giving relations for the class group on prime ideal generators
(𝔭_{i})_{1 ≤ i ≤ r}.

*bnf*[2] contains the matrix B, i.e. the matrix containing the
expressions of the prime ideal factorbase in terms of the 𝔭_{i}.
It is an r x c matrix.

*bnf*[3] contains the complex logarithmic embeddings of the system of
fundamental units which has been found. It is an (r_{1}+r_{2}) x (r_{1}+r_{2}-1)
matrix.

*bnf*[4] contains the matrix M"_{C} of Archimedean components of the
relations of the matrix (W|B).

*bnf*[5] contains the prime factor base, i.e. the list of prime
ideals used in finding the relations.

*bnf*[6] contains a dummy 0.

*bnf*[7] or

is equal to the number field data
*bnf*.nf*nf* as would be given by `nfinit`

.

*bnf*[8] is a vector containing the classgroup

as a finite abelian group, the regulator *bnf*.clgp

,
the number of roots of unity and a generator *bnf*.reg

, the
fundamental units *bnf*.tu*in expanded form*

. If the
fundamental units were omitted in the *bnf*.fu*bnf*,

returns
the sentinel value 0. If *bnf*.fu*flag* = 1, this vector contain also algebraic
data corresponding to the fundamental units and to the discrete logarithm
problem (see `bnfisprincipal`

). In particular, if *flag* = 1 we may
*only* know the units in factored form: the first call to

expands them, which may be very costly, then caches the
result.*bnf*.fu

*bnf*[9] is a vector used in `bnfisprincipal`

only
and obtained as follows. Let D = U W V obtained by applying the
Smith normal form algorithm to the matrix W ( = *bnf*[1]) and
let U_{r} be the reduction of U modulo D. The first elements of the
factorbase are given (in terms of `bnf.gen`

) by the columns of U_{r},
with Archimedean component g_{a}; let also GD_{a} be the Archimedean
components of the generators of the (principal) ideals defined by the
`bnf.gen[i]^bnf.cyc[i]`

. Then *bnf*[9] = [U_{r}, g_{a}, GD_{a}],
followed by technical exact components which allow to recompute g_{a} and
GD_{a} to higher accuracy.

*bnf*[10] is by default unused and set equal to 0. This field is used
to store further information about the field as it becomes available, which
is rarely needed, hence would be too expensive to compute during the initial
`bnfinit`

call. For instance, the generators of the principal ideals
`bnf.gen[i]^bnf.cyc[i]`

(during a call to `bnrisprincipal`

), or
those corresponding to the relations in W and B (when the `bnf`

internal precision needs to be increased).

The library syntax is `GEN `

.**bnfinit0**(GEN P, long flag, GEN tech = NULL, long prec)

Also available is `GEN `

,
corresponding to **Buchall**(GEN P, long flag, long prec)`tech = NULL`

, where
`flag`

is either 0 (default) or `nf_FORCE`

(include all data in
algebraic form). The function
`GEN `

gives direct access to the technical parameters.**Buchall_param**(GEN P, double c1, double c2, long nrpid, long flag, long prec)

Computes a complete system of
solutions (modulo units of positive norm) of the absolute norm equation
Norm(a) = x,
where a is an integer in *bnf*. If *bnf* has not been certified,
the correctness of the result depends on the validity of GRH.

See also `bnfisnorm`

.

The library syntax is `GEN `

.
The function **bnfisintnorm**(GEN bnf, GEN x)`GEN `

,
where **bnfisintnormabs**(GEN bnf, GEN a)`bnf`

is a true *bnf* structure,
returns a complete system of solutions modulo units of the absolute norm
equation |Norm(x) |= |a|. As fast as `bnfisintnorm`

, but solves
the two equations Norm(x) = ± a simultaneously.

Tries to tell whether the
rational number x is the norm of some element y in *bnf*. Returns a
vector [a,b] where x = Norm(a)*b. Looks for a solution which is an S-unit,
with S a certain set of prime ideals containing (among others) all primes
dividing x. If *bnf* is known to be Galois, you may set *flag* = 0
(in this case, x is a norm iff b = 1). If *flag* is nonzero the program adds
to S the following prime ideals, depending on the sign of *flag*. If *flag* > 0,
the ideals of norm less than *flag*. And if *flag* < 0 the ideals dividing *flag*.

Assuming GRH, the answer is guaranteed (i.e. x is a norm iff b = 1),
if S contains all primes less than 12log(disc(*Bnf*))^2, where
*Bnf* is the Galois closure of *bnf*.

See also `bnfisintnorm`

.

The library syntax is `GEN `

.**bnfisnorm**(GEN bnf, GEN x, long flag)

*bnf* being the
number field data output by `bnfinit`

, and x being an ideal, this
function tests whether the ideal is principal or not. The result is more
complete than a simple true/false answer and solves a general discrete
logarithm problem. Assume the class group is ⨁ (ℤ/d_{i}ℤ)g_{i}
(where the generators g_{i} and their orders d_{i} are respectively given by
`bnf.gen`

and `bnf.cyc`

). The routine returns a row vector [e,t],
where e is a vector of exponents 0 ≤ e_{i} < d_{i}, and t is a number
field element such that
x = (t) ∏_{i} g_{i}^{ei}.
For *given* g_{i} (i.e. for a given `bnf`

), the e_{i} are unique,
and t is unique modulo units.

In particular, x is principal if and only if e is the zero vector. Note that the empty vector, which is returned when the class number is 1, is considered to be a zero vector (of dimension 0).

? K = bnfinit(y^2+23); ? K.cyc %2 = [3] ? K.gen %3 = [[2, 0; 0, 1]] \\ a prime ideal above 2 ? P = idealprimedec(K,3)[1]; \\ a prime ideal above 3 ? v = bnfisprincipal(K, P) %5 = [[2]~, [3/4, 1/4]~] ? idealmul(K, v[2], idealfactorback(K, K.gen, v[1])) %6 = [3 0] [0 1] ? % == idealhnf(K, P) %7 = 1

The binary digits of *flag* mean:

***** 1: If set, outputs [e,t] as explained above, otherwise returns
only e, which is easier to compute. The following idiom only tests
whether an ideal is principal:

is_principal(bnf, x) = !bnfisprincipal(bnf,x,0);

***** 2: It may not be possible to recover t, given the initial accuracy
to which the `bnf`

structure was computed. In that case, a warning is
printed and t is set equal to the empty vector `[]~`

. If this bit is
set, increase the precision and recompute needed quantities until t can be
computed. Warning: setting this may induce *lengthy* computations, and
the result may be too large to be physically representable in any case.
You should consider using flag 4 instead.

***** 4: Return t in factored form (compact representation),
as a small product of S-units for a small set of finite places S,
possibly with huge exponents. This kind of result can be cheaply mapped to
K^{*}/(K^{*})^ℓ or to ℂ or ℚ_{p} to bounded accuracy and this is usually
enough for applications. Explicitly expanding such a compact representation
is possible using `nffactorback`

but may be *very* costly.
The algorithm is guaranteed to succeed if the `bnf`

was computed using
`bnfinit(,1)`

. If not, the algorithm may fail to compute a huge
generator in this case (and replace it by `[]~`

). This is orders of
magnitude faster than flag 2 when the generators are indeed large.

The library syntax is `GEN `

.
Instead of the above hardcoded numerical flags, one should
rather use an or-ed combination of the symbolic flags **bnfisprincipal0**(GEN bnf, GEN x, long flag)`nf_GEN`

(include
generators, possibly a place holder if too difficult), `nf_GENMAT`

(include generators in compact form) and
`nf_FORCE`

(insist on finding the generators, a no-op if `nf_GENMAT`

is included).

This function is obsolete, use `bnfisunit`

.

The library syntax is `GEN `

.**bnfissunit**(GEN bnf, GEN sfu, GEN x)

*bnf* being the number field data
output by `bnfinit`

and x being an algebraic number (type integer,
rational or polmod), this outputs the decomposition of x on the fundamental
units and the roots of unity if x is a unit, the empty vector otherwise.
More precisely, if u_{1},...,u_{r} are the fundamental units, and ζ
is the generator of the group of roots of unity (`bnf.tu`

), the output is
a vector [x_{1},...,x_{r},x_{r+1}] such that x = u_{1}^{x1}...
u_{r}^{xr}.ζ^{xr+1}. The x_{i} are integers but the last one
(i = r+1) is only defined modulo the order w of ζ and is guaranteed
to be in [0,w[.

Note that *bnf* need not contain the fundamental units explicitly: it may
contain the placeholder 0 instead:

? setrand(1); bnf = bnfinit(x^2-x-100000); ? bnf.fu %2 = 0 ? u = [119836165644250789990462835950022871665178127611316131167, \ 379554884019013781006303254896369154068336082609238336]~; ? bnfisunit(bnf, u) %3 = [-1, 0]~

The given u is 1/u_{1}, where u_{1} is the fundamental
unit implicitly stored in *bnf*. In this case, u_{1} was not computed
and stored in algebraic form since the default accuracy was too low. Re-run
the `bnfinit`

command at `\g1`

or higher to see such diagnostics.

This function allows x to be given in factored form, but it then assumes that x is an actual unit. (Because it is general too costly to check whether this is the case.)

? { v = [2, 85; 5, -71; 13, -162; 17, -76; 23, -37; 29, -104; [224, 1]~, -66; [-86, 1]~, 86; [-241, 1]~, -20; [44, 1]~, 30; [124, 1]~, 11; [125, -1]~, -11; [-214, 1]~, 33; [-213, -1]~, -33; [189, 1]~, 74; [190, -1]~, 104; [-168, 1]~, 2; [-167, -1]~, -8]; } ? bnfisunit(bnf,v) %5 = [1, 0]~

Note that v is the fundamental unit of `bnf`

given in
compact (factored) form.

If the argument `U`

is present, as output by `bnfunits(bnf, S)`

,
then the function decomposes x on the S-units generators given in
`U[1]`

.

? bnf = bnfinit(x^4 - x^3 + 4*x^2 + 3*x + 9, 1); ? bnf.sign %2 = [0, 2] ? S = idealprimedec(bnf,5); #S %3 = 2 ? US = bnfunits(bnf,S); ? g = US[1]; #g \\ #S = #g, four S-units generators, in factored form %5 = 4 ? g[1] %6 = [[6, -3, -2, -2]~ 1] ? g[2] %7 = [[-1, 1/2, -1/2, -1/2]~ 1] [ [4, -2, -1, -1]~ 1] ? [nffactorback(bnf, x) | x <- g] %8 = [[6, -3, -2, -2]~, [-5, 5, 0, 0]~, [-1, 1, -1, 0]~, [1, -1, 0, 0]~] ? u = [10,-40,24,11]~; ? a = bnfisunit(bnf, u, US) %9 = [2, 0, 1, 4]~ ? nffactorback(bnf, g, a) \\ prod_{i}g[i]^a[i] still in factored form %10 = [[6, -3, -2, -2]~ 2] [ [0, 0, -1, -1]~ 1] [ [2, -1, -1, 0]~ -2] [ [1, 1, 0, 0]~ 2] [ [-1, 1, 1, 1]~ -1] [ [1, -1, 0, 0]~ 4] ? nffactorback(bnf,%) \\ u = prod_{i}g[i]^a[i] %11 = [10, -40, 24, 11]~

The library syntax is `GEN `

.
Also available is **bnfisunit0**(GEN bnf, GEN x, GEN U = NULL)`GEN `

for U =
**bnfisunit**(GEN bnf, GEN x)`NULL`

.

Let *bnf* be a *bnf* structure attached to the number field F and let l be
a prime number (hereafter denoted ℓ for typographical reasons). Return
the logarithmic ℓ-class group ~{Cl}_{F}
of F. This is an abelian group, conjecturally finite (known to be finite
if F/ℚ is abelian). The function returns if and only if
the group is indeed finite (otherwise it would run into an infinite loop).
Let S = { 𝔭_{1},..., 𝔭_{k}} be the set of ℓ-adic places
(maximal ideals containing ℓ).
The function returns [D, G(ℓ), G'], where

***** D is the vector of elementary divisors for ~{Cl}_{F}.

***** G(ℓ) is the vector of elementary divisors for
the (conjecturally finite) abelian group
~{Cl}(ℓ) =
{ 𝔞 = ∑_{i ≤ k} a_{i} 𝔭_{i} : deg_{F} 𝔞 = 0},
where the 𝔭_{i} are the ℓ-adic places of F; this is a
subgroup of ~{Cl}.

***** G' is the vector of elementary divisors for the ℓ-Sylow Cl'
of the S-class group of F; the group ~{Cl} maps to Cl'
with a simple co-kernel.

The library syntax is `GEN `

.**bnflog**(GEN bnf, GEN l)

Let *nf* be a *nf* structure attached to a number field F,
and let l be a prime number (hereafter
denoted ℓ). The
ℓ-adified group of id\`{e}les of F quotiented by
the group of logarithmic units is identified to the ℓ-group
of logarithmic divisors ⨁ ℤ_ℓ [𝔭], generated by the
maximal ideals of F.

The *degree* map deg_{F} is additive with values in ℤ_ℓ,
defined by deg_{F} 𝔭 = ~{f}_{𝔭} deg_ℓ p,
where the integer ~{f}_{𝔭} is as in `bnflogef`

and deg_ℓ p
is log_ℓ p for p != ℓ, log_ℓ (1 + ℓ) for
p = ℓ != 2 and log_ℓ (1 + 2^2) for p = ℓ = 2.

Let A = ∏ 𝔭^{n𝔭} be an ideal and let ~{A} =
∑ n_𝔭 [𝔭] be the attached logarithmic divisor. Return the
exponential of the ℓ-adic logarithmic degree deg_{F} A, which is a
natural number.

The library syntax is `GEN `

.**bnflogdegree**(GEN nf, GEN A, GEN l)

Let *nf* be a *nf* structure attached to a number field F
and let *pr* be a *prid* structure attached to a
maximal ideal 𝔭 / p. Return
[~{e}(F_𝔭 / ℚ_{p}), ~{f}(F_𝔭 / ℚ_{p})]
the logarithmic ramification and residue degrees. Let ℚ_{p}^c/ℚ_{p} be the
cyclotomic ℤ_{p}-extension, then
~{e} = [F_𝔭 : F_𝔭 ∩ ℚ_{p}^c] and
~{f} = [F_𝔭 ∩ ℚ_{p}^c : ℚ_{p}]. Note that
~{e}~{f} = e(𝔭/p) f(𝔭/p), where e(𝔭/p) and f(𝔭/p) denote the
usual ramification and residue degrees.

? F = nfinit(y^6 - 3*y^5 + 5*y^3 - 3*y + 1); ? bnflogef(F, idealprimedec(F,2)[1]) %2 = [6, 1] ? bnflogef(F, idealprimedec(F,5)[1]) %3 = [1, 2]

The library syntax is `GEN `

.**bnflogef**(GEN nf, GEN pr)

*bnf* being as output by
`bnfinit`

, computes the narrow class group of *bnf*. The output is
a 3-component row vector v analogous to the corresponding class group
component

: the first component
is the narrow class number *bnf*.clgp`v.no`

, the second component is a vector
containing the SNF cyclic components `v.cyc`

of
the narrow class group, and the third is a vector giving the generators of
the corresponding `v.gen`

cyclic groups. Note that this function is a
special case of `bnrinit`

; the *bnf* need not contain fundamental
units.

The library syntax is `GEN `

.**bnfnarrow**(GEN bnf)

*bnf* being as output by
`bnfinit`

, this computes an r_{1} x (r_{1}+r_{2}-1) matrix having ±1
components, giving the signs of the real embeddings of the fundamental units.
The following functions compute generators for the totally positive units:

/* exponents of totally positive units generators on K.tu, K.fu */ tpuexpo(K)= { my(M, S = bnfsignunit(K), [m,n] = matsize(S)); \\ m = K.r1, n = r1+r2-1 S = matrix(m,n, i,j, if (S[i,j] < 0, 1,0)); S = concat(vectorv(m,i,1), S); \\ add sign(-1) M = matkermod(S, 2); if (M, mathnfmodid(M, 2), 2*matid(n+1)) } /* totally positive fundamental units of bnf K */ tpu(K)= { my(ex = tpuexpo(K)[,^1]); \\ remove ex[,1], corresponds to 1 or -1 my(v = concat(K.tu[2], K.fu)); [ nffactorback(K, v, c) | c <- ex]; }

The library syntax is `GEN `

.**signunits**(GEN bnf)

Computes the fundamental S-units of the
number field *bnf* (output by `bnfinit`

), where S is a list of
prime ideals (output by `idealprimedec`

). The output is a vector v with
6 components.

v[1] gives a minimal system of (integral) generators of the S-unit group modulo the unit group.

v[2] contains technical data needed by `bnfissunit`

.

v[3] is an obsoleted component, now the empty vector.

v[4] is the S-regulator (this is the product of the regulator, the S-class number and the natural logarithms of the norms of the ideals in S).

v[5] gives the S-class group structure, in the usual abelian group format: a vector whose three components give in order the S-class number, the cyclic components and the generators.

v[6] is a copy of S.

The library syntax is `GEN `

.
Also available is
**bnfsunit**(GEN bnf, GEN S, long prec)`GEN `

which returns only v[1].**sunits_mod_units**(GEN bnf, GEN S)

Return the fundamental units of the number field
bnf output by bnfinit; if S is present and is a list of prime ideals,
compute fundamental S-units instead. The first component of the result
contains independent integral S-units generators: first nonunits, then
r_{1}+r_{2}-1 fundamental units, then the torsion unit. The result may be used
as an optional argument to bnfisunit. The units are given in compact form:
no expensive computation is attempted if the *bnf* does not already
contain units.

? bnf = bnfinit(x^4 - x^3 + 4*x^2 + 3*x + 9, 1); ? bnf.sign \\ r1 + r2 - 1 = 1 %2 = [0, 2] ? U = bnfunits(bnf); u = U[1]; ? #u \\ r1 + r2 = 2 units %5 = 2; ? u[1] \\ fundamental unit as factorization matrix %6 = [[0, 0, -1, -1]~ 1] [[2, -1, -1, 0]~ -2] [ [1, 1, 0, 0]~ 2] [ [-1, 1, 1, 1]~ -1] ? u[2] \\ torsion unit as factorization matrix %7 = [[1, -1, 0, 0]~ 1] ? [nffactorback(bnf, z) | z <- u] \\ same units in expanded form %8 = [[-1, 1, -1, 0]~, [1, -1, 0, 0]~]

Now an example involving S-units for a nontrivial S:

? S = idealprimedec(bnf,5); #S %9 = 2 ? US = bnfunits(bnf, S); uS = US[1]; ? g = [nffactorback(bnf, z) | z <- uS] \\ now 4 units %11 = [[6, -3, -2, -2]~, [-5, 5, 0, 0]~, [-1, 1, -1, 0]~, [1, -1, 0, 0]~] ? bnfisunit(bnf,[10,-40,24,11]~) %12 = []~ \\ not a unit ? e = bnfisunit(bnf, [10,-40,24,11]~, US) %13 = [2, 0, 1, 4]~ \\ ...but an S-unit ? nffactorback(bnf, g, e) %14 = [10, -40, 24, 11]~ ? nffactorback(bnf, uS, e) \\ in factored form %15 = [[6, -3, -2, -2]~ 2] [ [0, 0, -1, -1]~ 1] [ [2, -1, -1, 0]~ -2] [ [1, 1, 0, 0]~ 2] [ [-1, 1, 1, 1]~ -1] [ [1, -1, 0, 0]~ 4]

Note that in more complicated cases, any `nffactorback`

fully expanding an element in factored form could be *very* expensive.
On the other hand, the final example expands a factorization whose components
are themselves in factored form, hence the result is a factored form:
this is a cheap operation.

The library syntax is `GEN `

.**bnfunits**(GEN bnf, GEN S = NULL)

Let *bnr* be the number field data output by `bnrinit`

and
*H* be a square matrix defining a congruence subgroup of the
ray class group corresponding to *bnr* (the trivial congruence subgroup
if omitted). This function returns, for each character χ of the ray
class group which is trivial on H, the value at s = 1 (or s = 0) of the
abelian L-function attached to χ. For the value at s = 0, the
function returns in fact for each χ a vector [r_χ, c_χ] where
L(s, χ) = c.s^r + O(s^{r + 1})
near 0.

The argument *flag* is optional, its binary digits
mean 1: compute at s = 0 if unset or s = 1 if set, 2: compute the
primitive L-function attached to χ if unset or the L-function
with Euler factors at prime ideals dividing the modulus of *bnr* removed
if set (that is L_{S}(s, χ), where S is the
set of infinite places of the number field together with the finite prime
ideals dividing the modulus of *bnr*), 3: return also the character if
set.

K = bnfinit(x^2-229); bnr = bnrinit(K,1); bnrL1(bnr)

returns the order and the first nonzero term of L(s, χ) at s = 0 where χ runs through the characters of the class group of K = ℚ(sqrt{229}). Then

bnr2 = bnrinit(K,2); bnrL1(bnr2,,2)

returns the order and the first nonzero terms of L_{S}(s, χ) at s = 0
where χ runs through the characters of the class group of K and S is
the set of infinite places of K together with the finite prime 2. Note
that the ray class group modulo 2 is in fact the class group, so
`bnrL1(bnr2,0)`

returns the same answer as `bnrL1(bnr,0)`

.

This function will fail with the message

*** bnrL1: overflow in zeta_get_N0 [need too many primes].

if the approximate functional equation requires us to sum too many terms (if the discriminant of K is too large).

The library syntax is `GEN `

.**bnrL1**(GEN bnr, GEN H = NULL, long flag, long prec)

Returns all characters χ on G such that
χ(g_{i}) = e(v_{i}), where e(x) = exp(2iπ x). G is allowed to be a
*bnr* struct (representing a ray class group) or a *znstar*
(representing (ℤ/Nℤ)^{*}). If v is omitted,
returns all characters that are trivial on the g_{i}. Else the vectors g
and v must have the same length, the g_{i} must be elements of G, and
each v_{i} is a rational number whose denominator must divide the order of
g_{i} in G.

For convenience, the vector of the g_{i}
can be replaced by a matrix whose columns give their discrete logarithm
in G, for instance as given by `bnrisprincipal`

if G is a *bnr*;
in this particular case, G can be any finite abelian group
given by a vector of elementary divisors.

? G = bnrinit(bnfinit(x), [160,[1]], 1); /* (Z/160Z)^{*}*/ ? G.cyc %2 = [8, 4, 2] ? g = G.gen; ? bnrchar(G, g, [1/2,0,0]) %4 = [[4, 0, 0]] \\ a unique character ? bnrchar(G, [g[1],g[3]]) \\ all characters trivial on g[1] and g[3] %5 = [[0, 1, 0], [0, 2, 0], [0, 3, 0], [0, 0, 0]] ? bnrchar(G, [1,0,0;0,1,0;0,0,2]) %6 = [[0, 0, 1], [0, 0, 0]] \\ characters trivial on given subgroup ? G = znstar(75, 1); ? bnrchar(G, [2, 7], [11/20, 1/4]) %8 = [[1, 1]] \\ Dirichlet char: chi(2) = e(11/20), chi(7) = e(1/4)

The library syntax is `GEN `

.**bnrchar**(GEN G, GEN g, GEN v = NULL)

*bnr* being as output by `bnrinit`

, returns a relative equation
for the class field corresponding to the congruence group defined by
(*bnr*,*subgp*) (the full ray class field if *subgp* is
omitted). The subgroup can also be a `t_INT`

n,
meaning n.Cl_{f}. The function also handles a vector of
subgroup, e.g, from `subgrouplist`

and returns the vector of individual
results in this case.

If *flag* = 0, returns a vector of polynomials such that the compositum of the
corresponding fields is the class field; if *flag* = 1 returns a single
polynomial; if *flag* = 2 returns a single absolute polynomial.

? bnf = bnfinit(y^3+14*y-1); bnf.cyc %1 = [4, 2] ? pol = bnrclassfield(bnf,,1) \\ Hilbert class field %2 = x^8 - 2*x^7 + ... + Mod(11*y^2 - 82*y + 116, y^3 + 14*y - 1) ? rnfdisc(bnf,pol)[1] %3 = 1 ? bnr = bnrinit(bnf,3*5*7); bnr.cyc %4 = [24, 12, 12, 2] ? bnrclassfield(bnr,2) \\ maximal 2-elementary subextension %5 = [x^2 + (-21*y - 105), x^2 + (-5*y - 25), x^2 + (-y - 5), x^2 + (-y - 1)] \\ quadratic extensions of maximal conductor ? bnrclassfield(bnr, subgrouplist(bnr,[2])) %6 = [[x^2 - 105], [x^2 + (-105*y^2 - 1260)], [x^2 + (-105*y - 525)], [x^2 + (-105*y - 105)]] ? #bnrclassfield(bnr,subgrouplist(bnr,[2],1)) \\ all quadratic extensions %7 = 15

When the subgroup contains n Cl_{f}, where n is fixed,
it is advised to directly compute the `bnr`

modulo n to avoid expensive
discrete logarithms:

? bnf = bnfinit(y^2-5); p = 1594287814679644276013; ? bnr = bnrinit(bnf,p); \\ very slow time = 24,146 ms. ? bnrclassfield(bnr, 2) \\ ... even though the result is trivial %3 = [x^2 - 1594287814679644276013] ? bnr2 = bnrinit(bnf,p,,2); \\ now fast time = 1 ms. ? bnrclassfield(bnr2, 2) %5 = [x^2 - 1594287814679644276013]

This will save a lot of time when the modulus contains a maximal ideal whose residue field is large.

The library syntax is `GEN `

.**bnrclassfield**(GEN bnr, GEN subgp = NULL, long flag, long prec)

Let A, B, C define a class field L over a ground field K
(of type `[`

,
*bnr*]`[`

,
or *bnr*, *subgroup*]`[`

,
or *bnf*, *modulus*]`[`

,
Section se:CFT); this function returns the relative degree [L:K].*bnf*, *modulus*,*subgroup*]

In particular if A is a *bnf* (with units), and B a modulus,
this function returns the corresponding ray class number modulo B.
One can input the attached *bid* (with generators if the subgroup
C is non trivial) for B instead of the module itself, saving some time.

This function is faster than `bnrinit`

and should be used if only the
ray class number is desired. See `bnrclassnolist`

if you need ray class
numbers for all moduli less than some bound.

The library syntax is `GEN `

.
Also available is
**bnrclassno0**(GEN A, GEN B = NULL, GEN C = NULL)`GEN `

to compute the ray class number
modulo f.**bnrclassno**(GEN bnf,GEN f)

*bnf* being as
output by `bnfinit`

, and *list* being a list of moduli (with units) as
output by `ideallist`

or `ideallistarch`

, outputs the list of the
class numbers of the corresponding ray class groups. To compute a single
class number, `bnrclassno`

is more efficient.

? bnf = bnfinit(x^2 - 2); ? L = ideallist(bnf, 100, 2); ? H = bnrclassnolist(bnf, L); ? H[98] %4 = [1, 3, 1] ? l = L[1][98]; ids = vector(#l, i, l[i].mod[1]) %5 = [[98, 88; 0, 1], [14, 0; 0, 7], [98, 10; 0, 1]]

The weird `l[i].mod[1]`

, is the first component of `l[i].mod`

, i.e.
the finite part of the conductor. (This is cosmetic: since by construction
the Archimedean part is trivial, I do not want to see it). This tells us that
the ray class groups modulo the ideals of norm 98 (printed as `%5`

) have
respectively order 1, 3 and 1. Indeed, we may check directly:

? bnrclassno(bnf, ids[2]) %6 = 3

The library syntax is `GEN `

.**bnrclassnolist**(GEN bnf, GEN list)

Given two abelian extensions A = `[bnr1, H1]`

and
B = `[bnr2, H2]`

, where `bnr1`

and `bnr2`

are two `bnr`

structures attached to the same base field, return their compositum
as `[bnr, H]`

. The modulus attached to `bnr`

need not be the
conductor of the compositum.

? Q = bnfinit(y); ? bnr1 = bnrinit(Q, [7, [1]]); bnr1.cyc %2 = [6] ? bnr2 = bnrinit(Q, [13, [1]]); bnr2.cyc %3 = [12] ? H1 = Mat(2); bnrclassfield(bnr1, H1) %4 = [x^2 + 7] ? H2 = Mat(2); bnrclassfield(bnr2, H2) %5 = [x^2 - 13] ? [bnr,H] = bnrcompositum([bnr1, H1], [bnr2,H2]); ? bnrclassfield(bnr,H) %7 = [x^2 - 13, x^2 + 7]

The library syntax is `GEN `

.**bnrcompositum**(GEN A, GEN B)

Conductor f of the subfield of a ray class field as defined by [A,B,C]
(of type `[`

,
*bnr*]`[`

,
*bnr*, *subgroup*]`[`

or
*bnf*, *modulus*]`[`

,
Section se:CFT)*bnf*, *modulus*, *subgroup*]

If *flag* = 0, returns f.

If *flag* = 1, returns [f, Cl_{f}, H], where Cl_{f} is the ray class group
modulo f, as a finite abelian group; finally H is the subgroup of Cl_{f}
defining the extension.

If *flag* = 2, returns [f, *bnr*(f), H], as above except Cl_{f} is
replaced by a `bnr`

structure, as output by `bnrinit`

(,f), without
generators unless the input contained a *bnr* with generators.

In place of a subgroup H, this function also accepts a character
`chi`

= (a_{j}), expressed as usual in terms of the generators
`bnr.gen`

: χ(g_{j}) = exp(2iπ a_{j} / d_{j}), where g_{j} has
order d_{j} = `bnr.cyc[j]`

. In which case, the function returns
respectively

If *flag* = 0, the conductor f of Ker χ.

If *flag* = 1, [f, Cl_{f}, χ_{f}], where χ_{f} is χ expressed
on the minimal ray class group, whose modulus is the conductor.

If *flag* = 2, [f, *bnr*(f), χ_{f}].

**Note.** Using this function with *flag* != 0 is usually a
bad idea and kept for compatibility and convenience only: *flag* = 1 has
always been useless, since it is no faster than *flag* = 2 and returns less
information; *flag* = 2 is mostly OK with two subtle drawbacks:

***** it returns the full *bnr* attached to the full ray class
group, whereas in applications we only need Cl_{f} modulo N-th powers,
where N is any multiple of the exponent of Cl_{f}/H. Computing directly the
conductor, then calling `bnrinit`

with optional argument N avoids this
problem.

***** computing the *bnr* needs only be done once for each
conductor, which is not possible using this function.

For maximal efficiency, the recommended procedure is as follows. Starting
from data (character or congruence subgroups) attached to a modulus m,
we can first compute the conductors using this function with default *flag* =
0. Then for all data with a common conductor f | m, compute (once!) the
*bnr* attached to f using `bnrinit`

(modulo N-th powers for
a suitable N!) and finally map original data to the new *bnr* using
`bnrmap`

.

The library syntax is `GEN `

.**bnrconductor0**(GEN A, GEN B = NULL, GEN C = NULL, long flag)

Also available are `GEN `

and **bnrconductor**(GEN bnr, GEN H, long flag)`GEN `

which returns ray class groups modulo **bnrconductormod**(GEN bnr, GEN H, long flag, GEN cycmod)`cycmod`

-th powers.

This function is obsolete, use *bnrconductor*.

The library syntax is `GEN `

.**bnrconductorofchar**(GEN bnr, GEN chi)

A, B, C defining a class field L over a ground field K
(of type `[`

,
*bnr*]`[`

,
*bnr*, *subgroup*]`[`

,
*bnr*, *character*]`[`

or
*bnf*, *modulus*]`[`

,
Section se:CFT), outputs data [N,r*bnf*, *modulus*, *subgroup*]_{1},D] giving the discriminant and
signature of L, depending on the binary digits of *flag*:

***** 1: if this bit is unset, output absolute data related to L/ℚ:
N is the absolute degree [L:ℚ], r_{1} the number of real places of L,
and D the discriminant of L/ℚ. Otherwise, output relative data for L/K:
N is the relative degree [L:K], r_{1} is the number of real places of K
unramified in L (so that the number of real places of L is equal to r_{1}
times N), and D is the relative discriminant ideal of L/K.

***** 2: if this bit is set and if the modulus is not the conductor of L,
only return 0.

The library syntax is `GEN `

.**bnrdisc0**(GEN A, GEN B = NULL, GEN C = NULL, long flag)

*bnf* being as output by `bnfinit`

(with units), computes a
list of discriminants of Abelian extensions of the number field by increasing
modulus norm up to bound *bound*. The ramified Archimedean places are
given by *arch*; all possible values are taken if *arch* is omitted.

The alternative syntax `bnrdisclist`

(*bnf*,*list*) is
supported, where *list* is as output by `ideallist`

or
`ideallistarch`

(with units), in which case *arch* is disregarded.

The output v is a vector, where v[k] is itself a vector w, whose length is the number of ideals of norm k.

***** We consider first the case where *arch* was specified. Each
component of w corresponds to an ideal m of norm k, and
gives invariants attached to the ray class field L of *bnf* of
conductor [m, *arch*]. Namely, each contains a vector [m,d,r,D] with
the following meaning: m is the prime ideal factorization of the modulus,
d = [L:ℚ] is the absolute degree of L, r is the number of real places
of L, and D is the factorization of its absolute discriminant. We set d
= r = D = 0 if m is not the finite part of a conductor.

***** If *arch* was omitted, all t = 2^{r1} possible values are taken
and a component of w has the form
[m, [[d_{1},r_{1},D_{1}],..., [d_{t},r_{t},D_{t}]]],
where m is the finite part of the conductor as above, and
[d_{i},r_{i},D_{i}] are the invariants of the ray class field of conductor
[m,v_{i}], where v_{i} is the i-th Archimedean component, ordered by
inverse lexicographic order; so v_{1} = [0,...,0], v_{2} = [1,0...,0],
etc. Again, we set d_{i} = r_{i} = D_{i} = 0 if [m,v_{i}] is not a conductor.

Finally, each prime ideal pr = [p,α,e,f,β] in the prime factorization m is coded as the integer p.n^2+(f-1).n+(j-1), where n is the degree of the base field and j is such that

`pr = idealprimedec(`

.*nf*,p)[j]

m can be decoded using `bnfdecodemodule`

.

Note that to compute such data for a single field, either `bnrclassno`

or `bnrdisc`

are (much) more efficient.

The library syntax is `GEN `

.**bnrdisclist0**(GEN bnf, GEN bound, GEN arch = NULL)

Apply the automorphism given by its matrix *mat* to the congruence
subgroup H given as a HNF matrix.
The matrix *mat* can be computed with `bnrgaloismatrix`

.

The library syntax is `GEN `

.**bnrgaloisapply**(GEN bnr, GEN mat, GEN H)

Return the matrix of the action of the automorphism *aut* of the base
field `bnf.nf`

on the generators of the ray class field `bnr.gen`

.
The automorphism
*aut* can be given as a polynomial, an algebraic number, or a vector of
automorphisms and must stabilize the modulus `bnr.mod`

. We also
allow a Galois group as output by `galoisinit`

, in which case a
vector of matrices is returned corresponding to the generators
`aut.gen`

.
Note: This function only makes sense when the ray class field attached to
*bnr* is Galois, which is not checked.

The generators `bnr.gen`

need not be explicitly computed in the input
*bnr*, which saves time: the result is well defined in this case also.

? K = bnfinit(a^4-3*a^2+253009); B = bnrinit(K,9); B.cyc %1 = [8400, 12, 6, 3] ? G = nfgaloisconj(K) %2 = [-a, a, -1/503*a^3 + 3/503*a, 1/503*a^3 - 3/503*a]~ ? bnrgaloismatrix(B, G[2]) \\ G[2] = Id ... %3 = [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] ? bnrgaloismatrix(B, G[3]) \\ automorphism of order 2 %4 = [799 0 0 2800] [ 0 7 0 4] [ 4 0 5 2] [ 0 0 0 2] ? M = %^2; for (i=1, #B.cyc, M[i,] %= B.cyc[i]); M %5 = \\ acts on ray class group as automorphism of order 2 [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1]

See `bnrisgalois`

for further examples.

The library syntax is `GEN `

.
When aut is a polynomial or an algebraic number,
**bnrgaloismatrix**(GEN bnr, GEN aut)`GEN `

is available.**bnrautmatrix**(GEN bnr, GEN aut)

*bnf* is as
output by `bnfinit`

(including fundamental units), f is a modulus,
initializes data linked to the ray class group structure corresponding to
this module, a so-called `bnr`

structure. One can input the attached
*bid* with generators for f instead of the module itself, saving some
time. (As in `idealstar`

, the finite part of the conductor may be given
by a factorization into prime ideals, as produced by `idealfactor`

.)

If the positive integer `cycmod`

is present, only compute the ray class
group modulo `cycmod`

, which may save a lot of time when some maximal
ideals in f have a huge residue field. In applications, we are given
a congruence subgroup H and study the class field attached to
Cl_{f}/H. If that finite Abelian group has an exponent which divides
`cycmod`

, then we have changed nothing theoretically, while trivializing
expensive discrete logs in residue fields (since computations can be
made modulo `cycmod`

-th powers). This is useful in `bnrclassfield`

,
for instance when computing p-elementary extensions.

The following member functions are available
on the result: `.bnf`

is the underlying *bnf*,
`.mod`

the modulus, `.bid`

the `bid`

structure attached to the
modulus; finally, `.clgp`

, `.no`

, `.cyc`

, `.gen`

refer to the
ray class group (as a finite abelian group), its cardinality, its elementary
divisors, its generators (only computed if *flag* = 1).

The last group of functions are different from the members of the underlying
*bnf*, which refer to the class group; use

to access these, e.g. *bnr*.bnf.*xxx*

to get the cyclic decomposition
of the class group.*bnr*.bnf.cyc

They are also different from the members of the underlying *bid*, which
refer to (ℤ_{K}/f)^{*}; use

to access these,
e.g. *bnr*.bid.*xxx*

to get φ(f).*bnr*.bid.no

If *flag* = 0 (default), the generators of the ray class group are not
explicitly computed, which saves time. Hence

would
produce an error. Note that implicit generators are still fixed and stored
in the *bnr*.gen*bnr* (and guaranteed to be the same for fixed *bnf* and
*bid* inputs), in terms of `bnr.bnf.gen`

and `bnr.bid.gen`

.
The computation which is not performed is the expansion of such products
in the ray class group so as to fix eplicit ideal representatives.

If *flag* = 1, as the default, except that generators are computed.

The library syntax is `GEN `

.
Instead of the above hardcoded numerical flags, one should rather use
**bnrinitmod**(GEN bnf, GEN f, long flag, GEN cycmod = NULL)`GEN `

where an omitted **Buchraymod**(GEN bnf, GEN module, long flag, GEN cycmod)`cycmod`

is coded as `NULL`

and flag is an or-ed
combination of `nf_GEN`

(include generators) and `nf_INIT`

(if
omitted, return just the cardinality of the ray class group and its structure),
possibly 0. Or simply
`GEN `

when **Buchray**(GEN bnf, GEN module, long flag)`cycmod`

is `NULL`

.

Fast variant of `bnrconductor`

(A,B,C); A, B, C represent
an extension of the base field, given by class field theory
(see Section se:CFT). Outputs 1 if this modulus is the conductor, and 0
otherwise. This is slightly faster than `bnrconductor`

when the
character or subgroup is not primitive.

The library syntax is `long `

.**bnrisconductor0**(GEN A, GEN B = NULL, GEN C = NULL)

Check whether the class field attached to the subgroup H is Galois
over the subfield of `bnr.nf`

fixed by the group *gal*, which can be
given as output by `galoisinit`

, or as a matrix or a vector of matrices as
output by `bnrgaloismatrix`

, the second option being preferable, since it
saves the recomputation of the matrices. Note: The function assumes that the
ray class field attached to *bnr* is Galois, which is not checked.

In the following example, we lists the congruence subgroups of subextension of degree at most 3 of the ray class field of conductor 9 which are Galois over the rationals.

? K = bnfinit(a^4-3*a^2+253009); B = bnrinit(K,9); G = galoisinit(K); ? [H | H<-subgrouplist(B,3), bnrisgalois(B,G,H)]; time = 160 ms. ? M = bnrgaloismatrix(B,G); ? [H | H<-subgrouplist(B,3), bnrisgalois(B,M,H)] time = 1 ms.

The second computation is much faster since `bnrgaloismatrix(B,G)`

is
computed only once.

The library syntax is `long `

.**bnrisgalois**(GEN bnr, GEN gal, GEN H)

Let *bnr* be the ray class group data output by
`bnrinit`

(,,1) and let x be an ideal in any form, coprime
to the modulus f = `bnr.mod`

. Solves the discrete logarithm problem
in the ray class group, with respect to the generators `bnr.gen`

,
in a way similar to `bnfisprincipal`

. If x is not coprime to the
modulus of *bnr* the result is undefined. Note that *bnr* need not
contain the ray class group generators, i.e. it may be created with
`bnrinit`

(,,0); in that case, although `bnr.gen`

is undefined, we
can still fix natural generators for the ray class group (in terms of the
generators in `bnr.bnf.gen`

and `bnr.bid.gen`

) and compute with
respect to them.

The binary digits of *flag* (default *flag* = 1) mean:

***** 1: If set returns a 2-component vector [e,α] where e
is the vector of components of x on the ray class group generators,
α is an element congruent to 1 mod^{*} f such that
x = α ∏_{i} g_{i}^{ei}. If unset, returns only e.

***** 4: If set, returns [e,α] where α is given in factored
form (compact representation). This is orders of magnitude faster.

? K = bnfinit(x^2 - 30); bnr = bnrinit(K, [4, [1,1]]); ? bnr.clgp \\ ray class group is isomorphic to Z/4 x Z/2 x Z/2 %2 = [16, [4, 2, 2]] ? P = idealprimedec(K, 3)[1]; \\ the ramified prime ideal above 3 ? bnrisprincipal(bnr,P) \\ bnr.gen undefined ! %5 = [[3, 0, 0]~, 9] ? bnrisprincipal(bnr,P, 0) \\ omit principal part %5 = [3, 0, 0]~ ? bnr = bnrinit(bnr, bnr.bid, 1); \\ include explicit generators ? bnrisprincipal(bnr,P) \\ ... alpha is different ! %7 = [[3, 0, 0]~, 1/128625]

It may be surprising that the generator α is different
although the underlying *bnf* and *bid* are the same. This defines
unique generators for the ray class group as ideal *classes*, whether
we use `bnrinit(,0)`

or `bnrinit(,1)`

. But the actual ideal
representatives (implicit if the flag is 0, computed and stored in the
*bnr* if the flag is 1) are in general different and this is what
happens here. Indeed, the implicit generators are naturally expressed
in terms of `bnr.bnf.gen`

and `bnr.bid.gen`

and *then*
expanded and simplified (in the same ideal class) so that we obtain ideal
representatives for `bnr.gen`

which are as simple as possible.
And indeed the quotient of the two α found is 1 modulo the
conductor (and positive at the infinite places it contains), and this is the
only guaranteed property.

Beware that, when `bnr`

is generated using `bnrinit(, cycmod)`

, the
results are given in Cl_{f} modulo `cycmod`

-th powers:

? bnr2 = bnrinit(K, bnr.mod,, 2); \\ modulo squares ? bnr2.clgp %9 = [8, [2, 2, 2]] \\ bnr.clgp tensored by Z/2Z ? bnrisprincipal(bnr2,P, 0) %10 = [1, 0, 0]~

The library syntax is `GEN `

.
Instead of hardcoded numerical flags, one should rather use
**bnrisprincipal**(GEN bnr, GEN x, long flag)`GEN `

for **isprincipalray**(GEN bnr, GEN x)`flag`

= 0, and if you
want generators:

bnrisprincipal(bnr, x, nf_GEN)

Also available is
`GEN `

that returns the discrete logarithm of x modulo the **bnrisprincipalmod**(GEN bnr, GEN x, GEN mod, long flag)`t_INT`

`mod`

; the value `mod = NULL`

is treated as 0 (full discrete
logarithm), and `flag`

= 1 is not allowed if `mod`

is set.

This function has two different uses:

***** if A and B are *bnr* structures for the same *bnf* attached
to moduli m_{A} and m_{B} with m_{B} | m_{A}, return the canonical surjection
from A to B, i.e. from the ray class group moodulo m_{A} to the ray
class group modulo m_{B}. The map is coded by a triple
[M,*cyc*_{A},*cyc*_{B}]:
M gives the image of the fixed ray class group generators of A in
terms of the ones in B, *cyc*_{A} and *cyc*_{B} are the cyclic
structures `A.cyc`

and `B.cyc`

respectively. Note that this function
does *not* need A or B to contain explicit generators for the ray
class groups: they may be created using `bnrinit(,0)`

.

If B is only known modulo N-th powers (from `bnrinit(,N)`

), the result
is correct provided N is a multiple of the exponent of A.

***** if A is a projection map as above and B is either a congruence
subgroup H, or a ray class character χ, or a discrete logarithm
(from `bnrisprincipal`

) modulo m_{A} whose conductor
divides m_{B}, return the image of the subgroup (resp. the character, the
discrete logarighm) as defined modulo m_{B}. The main use of this variant is
to compute the primitive subgroup or character attached to a *bnr* modulo
their conductor. This is more efficient than `bnrconductor`

in two
respects: the *bnr* attached to the conductor need only be computed once
and, most importantly, the ray class group can be computed modulo N-th
powers, where N is a multiple of the exponent of Cl_{mA} / H (resp.
of the order of χ). Whereas `bnrconductor`

is specified to return a
*bnr* attached to the full ray class group, which may lead to untractable
discrete logarithms in the full ray class group instead of a tiny quotient.

The library syntax is `GEN `

.**bnrmap**(GEN A, GEN B)

If χ = *chi* is a
character over *bnr*, not necessarily primitive, let
L(s,χ) = ∑_{id} χ(id) N(id)^{-s} be the attached
Artin L-function. Returns the so-called Artin root number, i.e. the
complex number W(χ) of modulus 1 such that

Λ(1-s,χ) = W(χ) Λ(s,χ)

where Λ(s,χ) = A(χ)^{s/2}γ_χ(s) L(s,χ) is
the enlarged L-function attached to L.

You can set *flag* = 1 if the character is known to be primitive. Example:

bnf = bnfinit(x^2 - x - 57); bnr = bnrinit(bnf, [7,[1,1]]); bnrrootnumber(bnr, [2,1])

returns the root number of the character χ of
Cl_{7 oo 1 oo 2}(ℚ(sqrt{229})) defined by χ(g_{1}^ag_{2}^b)
= ζ_{1}^{2a}ζ_{2}^b. Here g_{1}, g_{2} are the generators of the
ray-class group given by `bnr.gen`

and ζ_{1} = e^{2iπ/N1},
ζ_{2} = e^{2iπ/N2} where N_{1}, N_{2} are the orders of g_{1} and
g_{2} respectively (N_{1} = 6 and N_{2} = 3 as `bnr.cyc`

readily tells us).

The library syntax is `GEN `

.**bnrrootnumber**(GEN bnr, GEN chi, long flag, long prec)

*bnr* being as output by `bnrinit`

, finds a relative equation
for the class field corresponding to the modulus in *bnr* and the given
congruence subgroup (as usual, omit *subgroup* if you want the whole ray
class group).

The main variable of *bnr* must not be x, and the ground field and the
class field must be totally real. When the base field is ℚ, the vastly
simpler `galoissubcyclo`

is used instead. Here is an example:

bnf = bnfinit(y^2 - 3); bnr = bnrinit(bnf, 5); bnrstark(bnr)

returns the ray class field of ℚ(sqrt{3}) modulo 5. Usually, one wants to apply to the result one of

rnfpolredbest(bnf, pol) \\ compute a reduced relative polynomial rnfpolredbest(bnf, pol, 2) \\ compute a reduced absolute polynomial

The routine uses Stark units and needs to find a suitable auxiliary
conductor, which may not exist when the class field is not cyclic over the
base. In this case `bnrstark`

is allowed to return a vector of
polynomials defining *independent* relative extensions, whose compositum
is the requested class field. We decided that it was useful to keep the
extra information thus made available, hence the user has to take the
compositum herself, see `nfcompositum`

.

Even if it exists, the auxiliary conductor may be so large that later
computations become unfeasible. (And of course, Stark's conjecture may simply
be wrong.) In case of difficulties, try `bnrclassfield`

:

? bnr = bnrinit(bnfinit(y^8-12*y^6+36*y^4-36*y^2+9,1), 2); ? bnrstark(bnr) *** at top-level: bnrstark(bnr) *** ^ — — — — - *** bnrstark: need 3919350809720744 coefficients in initzeta. *** Computation impossible. ? bnrclassfield(bnr) time = 20 ms. %2 = [x^2 + (-2/3*y^6 + 7*y^4 - 14*y^2 + 3)]

The library syntax is `GEN `

.**bnrstark**(GEN bnr, GEN subgroup = NULL, long prec)

Gives as a vector the first b
coefficients of the Dedekind zeta function of the number field *nf*
considered as a Dirichlet series.

The library syntax is `GEN `

.**dirzetak**(GEN nf, GEN b)

This function is obsolete, use `nffactor`

.

factorization of the univariate polynomial x
over the number field defined by the (univariate) polynomial t. x may
have coefficients in ℚ or in the number field. The algorithm reduces to
factorization over ℚ (Trager's trick). The direct approach of
`nffactor`

, which uses van Hoeij's method in a relative setting, is
in general faster.

The main variable of t must be of *lower* priority than that of x
(see Section se:priority). However if nonrational number field elements
occur (as polmods or polynomials) as coefficients of x, the variable of
these polmods *must* be the same as the main variable of t. For
example

? factornf(x^2 + Mod(y, y^2+1), y^2+1); ? factornf(x^2 + y, y^2+1); \\ these two are OK ? factornf(x^2 + Mod(z,z^2+1), y^2+1) *** at top-level: factornf(x^2+Mod(z,z *** ^ — — — — — — -- *** factornf: inconsistent data in rnf function. ? factornf(x^2 + z, y^2+1) *** at top-level: factornf(x^2+z,y^2+1 *** ^ — — — — — — -- *** factornf: incorrect variable in rnf function.

The library syntax is `GEN `

.**polfnf**(GEN x, GEN t)

Let G be the group attached to the `galoisinit`

structure *gal*, and
let χ be the character of some representation ρ of the group G,
where a polynomial variable is to be interpreted as an o-th root of 1.
For instance, if `[T,o] = galoischartable(gal)`

the characters
χ are input as the columns of `T`

.

Return the degree-1 character detρ as the list of det ρ(g),
where g runs through representatives of the conjugacy classes
in `galoisconjclasses(gal)`

, with the same ordering.

? P = x^5 - x^4 - 5*x^3 + 4*x^2 + 3*x - 1; ? polgalois(P) %2 = [10, 1, 1, "D(5) = 5:2"] ? K = nfsplitting(P); ? gal = galoisinit(K); \\ dihedral of order 10 ? [T,o] = galoischartable(gal); ? chi = T[,1]; \\ trivial character ? galoischardet(gal, chi, o) %7 = [1, 1, 1, 1]~ ? [galoischardet(gal, T[,i], o) | i <- [1..#T]] \\ all characters %8 = [[1, 1, 1, 1]~, [1, 1, -1, 1]~, [1, 1, -1, 1]~, [1, 1, -1, 1]~]

The library syntax is `GEN `

.**galoischardet**(GEN gal, GEN chi, long o)

Let G be the group attached to the `galoisinit`

structure *gal*, and
let χ be the character of some representation ρ of the group
G, where a polynomial variable is to be interpreted as an o-th root of
1, e.g., if `[T,o] = galoischartable(gal)`

and χ is a column of
`T`

.
Return the list of characteristic polynomials det(1 - ρ(g)T),
where g runs through representatives of the conjugacy classes
in `galoisconjclasses(gal)`

, with the same ordering.

? T = x^5 - x^4 - 5*x^3 + 4*x^2 + 3*x - 1; ? polgalois(T) %2 = [10, 1, 1, "D(5) = 5:2"] ? K = nfsplitting(T); ? gal = galoisinit(K); \\ dihedral of order 10 ? [T,o] = galoischartable(gal); ? o %5 = 5 ? galoischarpoly(gal, T[,1], o) \\ T[,1] is the trivial character %6 = [-x + 1, -x + 1, -x + 1, -x + 1]~ ? galoischarpoly(gal, T[,3], o) %7 = [x^2 - 2*x + 1, x^2 + (y^3 + y^2 + 1)*x + 1, -x^2 + 1, x^2 + (-y^3 - y^2)*x + 1]~

The library syntax is `GEN `

.**galoischarpoly**(GEN gal, GEN chi, long o)

Compute the character table of G, where G is the underlying group of
the `galoisinit`

structure *gal*. The input *gal* is also allowed
to be a `t_VEC`

of permutations that is closed under products.
Let N be the number of conjugacy classes of G.
Return a `t_VEC`

[M,*e*] where e ≥ 1 is an integer
and M is a square `t_MAT`

of size N giving the character table
of G.

***** Each column corresponds to an irreducible character; the characters
are ordered by increasing dimension and the first column is the trivial
character (hence contains only 1's).

***** Each row corresponds to a conjugacy class; the conjugacy classes are
ordered as specified by `galoisconjclasses(gal)`

, in particular the
first row corresponds to the identity and gives the dimension χ(1)
of the irreducible representation attached to the successive characters
χ.

The value M[i,j] of the character j at the conjugacy class i
is represented by a polynomial in `y`

whose variable should be
interpreted as an e-th root of unity, i.e. as the lift of

Mod(y, polcyclo(e,'y))

(Note that M is the transpose of the usual orientation for character tables.)

The integer e divides the exponent of the group G and is chosen as small
as posible; for instance e = 1 when the characters are all defined over
ℚ, as is the case for S_{n}. Examples:

? K = nfsplitting(x^4+x+1); ? gal = galoisinit(K); ? [M,e] = galoischartable(gal); ? M~ \\ take the transpose to get the usual orientation %4 = [1 1 1 1 1] [1 -1 -1 1 1] [2 0 0 -1 2] [3 -1 1 0 -1] [3 1 -1 0 -1] ? e %5 = 1 ? {G = [Vecsmall([1, 2, 3, 4, 5]), Vecsmall([1, 5, 4, 3, 2]), Vecsmall([2, 1, 5, 4, 3]), Vecsmall([2, 3, 4, 5, 1]), Vecsmall([3, 2, 1, 5, 4]), Vecsmall([3, 4, 5, 1, 2]), Vecsmall([4, 3, 2, 1, 5]), Vecsmall([4, 5, 1, 2, 3]), Vecsmall([5, 1, 2, 3, 4]), Vecsmall([5, 4, 3, 2, 1])];} \\G = D10 ? [M,e] = galoischartable(G); ? M~ %8 = [1 1 1 1] [1 -1 1 1] [2 0 -y^3 - y^2 - 1 y^3 + y^2] [2 0 y^3 + y^2 -y^3 - y^2 - 1] ? e %9 = 5

The library syntax is `GEN `

.**galoischartable**(GEN gal)

*gal* being output by `galoisinit`

,
return the list of conjugacy classes of the underlying group.
The ordering of the classes is consistent with `galoischartable`

and the trivial class comes first.

? G = galoisinit(x^6+108); ? galoisidentify(G) %2 = [6, 1] \\ S_{3}? S = galoisconjclasses(G) %3 = [[Vecsmall([1,2,3,4,5,6])], [Vecsmall([3,1,2,6,4,5]),Vecsmall([2,3,1,5,6,4])], [Vecsmall([6,5,4,3,2,1]),Vecsmall([5,4,6,2,1,3]), Vecsmall([4,6,5,1,3,2])]] ? [[permorder(c[1]),#c] | c <- S ] %4 = [[1,1], [3,2], [2,3]]

This command also accepts subgroups returned by `galoissubgroups`

:

? subs = galoissubgroups(G); H = subs[5]; ? galoisidentify(H) %2 = [2, 1] \\ Z/2 ? S = galoisconjclasses(subgroups_of_{G}[5]); ? [[permorder(c[1]),#c] | c <- S ] %4 = [[1,1], [2,1]]

The library syntax is `GEN `

.**galoisconjclasses**(GEN gal)

*gal* being be a Galois group as output by `galoisinit`

,
export the underlying permutation group as a string suitable
for (no flags or *flag* = 0) GAP or (*flag* = 1) Magma. The following example
compute the index of the underlying abstract group in the GAP library:

? G = galoisinit(x^6+108); ? s = galoisexport(G) %2 = "Group((1, 2, 3)(4, 5, 6), (1, 4)(2, 6)(3, 5))" ? extern("echo \"IdGroup("s");\" | gap -q") %3 = [6, 1] ? galoisidentify(G) %4 = [6, 1]

This command also accepts subgroups returned by `galoissubgroups`

.

To *import* a GAP permutation into gp (for `galoissubfields`

for
instance), the following GAP function may be useful:

PermToGP := function(p, n) return Permuted([1..n],p); end; gap> p:= (1,26)(2,5)(3,17)(4,32)(6,9)(7,11)(8,24)(10,13)(12,15)(14,27) (16,22)(18,28)(19,20)(21,29)(23,31)(25,30) gap> PermToGP(p,32); [ 26, 5, 17, 32, 2, 9, 11, 24, 6, 13, 7, 15, 10, 27, 12, 22, 3, 28, 20, 19, 29, 16, 31, 8, 30, 1, 14, 18, 21, 25, 23, 4 ]

The library syntax is `GEN `

.**galoisexport**(GEN gal, long flag)

*gal* being be a Galois group as output by `galoisinit`

and
*perm* an element of *gal*.group, a vector of such elements
or a subgroup of *gal* as returned by galoissubgroups,
computes the fixed field of *gal* by the automorphism defined by the
permutations *perm* of the roots *gal*.roots. P is guaranteed to
be squarefree modulo *gal*.p.

If no flags or *flag* = 0, output format is the same as for `nfsubfield`

,
returning [P,x] such that P is a polynomial defining the fixed field, and
x is a root of P expressed as a polmod in *gal*.pol.

If *flag* = 1 return only the polynomial P.

If *flag* = 2 return [P,x,F] where P and x are as above and F is the
factorization of *gal*.pol over the field defined by P, where
variable v (y by default) stands for a root of P. The priority of v
must be less than the priority of the variable of *gal*.pol (see
Section se:priority).
In this case, P is also expressed in the variable v for compatibility
with F. Example:

? G = galoisinit(x^4+1); ? galoisfixedfield(G,G.group[2],2) %2 = [y^2 - 2, Mod(- x^3 + x, x^4 + 1), [x^2 - y*x + 1, x^2 + y*x + 1]]

computes the factorization x^4+1 = (x^2-sqrt{2}x+1)(x^2+sqrt{2}x+1)

The library syntax is `GEN `

where **galoisfixedfield**(GEN gal, GEN perm, long flag, long v = -1)`v`

is a variable number.

Query the `galpol`

package for a group of order a with index b
in the GAP4 Small Group library, by Hans Ulrich Besche, Bettina Eick and
Eamonn O'Brien.

The current version of `galpol`

supports groups of order a ≤ 143.
If b is omitted, return the number of isomorphism classes of
groups of order a.

The library syntax is `GEN `

.
Also available is **galoisgetgroup**(long a, long b)`GEN `

when b
is omitted.**galoisnbpol**(long a)

Query the `galpol`

package for a string describing the group of order
a with index b in the GAP4 Small Group library, by Hans Ulrich Besche,
Bettina Eick and Eamonn O'Brien.
The strings were generated using the GAP4 function `StructureDescription`

.
The command below outputs the names of all abstract groups of order 12:

? o = 12; N = galoisgetgroup(o); \\ # of abstract groups of order 12 ? for(i=1, N, print(i, ". ", galoisgetname(o,i))) 1. C3 : C4 2. C12 3. A4 4. D12 5. C6 x C2

The current version of `galpol`

supports groups of order a ≤ 143.
For a ≥ 16, it is possible for different groups to have the same name:

? o = 20; N = galoisgetgroup(o); ? for(i=1, N, print(i, ". ", galoisgetname(o,i))) 1. C5 : C4 2. C20 3. C5 : C4 4. D20 5. C10 x C2

The library syntax is `GEN `

.**galoisgetname**(long a, long b)

Query the `galpol`

package for a polynomial with Galois group
isomorphic to
GAP4(a,b), totally real if s = 1 (default) and totally complex if s = 2.
The current version of `galpol`

supports groups of order a ≤ 143.
The output is a vector [`pol`

, `den`

] where

***** `pol`

is the polynomial of degree a

***** `den`

is the denominator of `nfgaloisconj(pol)`

.
Pass it as an optional argument to `galoisinit`

or `nfgaloisconj`

to
speed them up:

? [pol,den] = galoisgetpol(64,4,1); ? G = galoisinit(pol); time = 352ms ? galoisinit(pol, den); \\ passing 'den' speeds up the computation time = 264ms ? % == %` %4 = 1 \\ same answer

If b and s are omitted, return the number of isomorphism classes of groups of order a.

The library syntax is `GEN `

.
Also available is **galoisgetpol**(long a, long b, long s)`GEN `

when b and s
are omitted.**galoisnbpol**(long a)

*gal* being be a Galois group as output by `galoisinit`

,
output the isomorphism class of the underlying abstract group as a
two-components vector [o,i], where o is the group order, and i is the
group index in the GAP4 Small Group library, by Hans Ulrich Besche, Bettina
Eick and Eamonn O'Brien.

This command also accepts subgroups returned by `galoissubgroups`

.

The current implementation is limited to degree less or equal to 127. Some larger "easy" orders are also supported.

The output is similar to the output of the function `IdGroup`

in GAP4.
Note that GAP4 `IdGroup`

handles all groups of order less than 2000
except 1024, so you can use `galoisexport`

and GAP4 to identify large
Galois groups.

The library syntax is `GEN `

.**galoisidentify**(GEN gal)

Computes the Galois group
and all necessary information for computing the fixed fields of the
Galois extension K/ℚ where K is the number field defined by
*pol* (monic irreducible polynomial in ℤ[X] or
a number field as output by `nfinit`

). The extension K/ℚ must be
Galois with Galois group "weakly" super-solvable, see below;
returns 0 otherwise. Hence this permits to quickly check whether a polynomial
of order strictly less than 48 is Galois or not.

The algorithm used is an improved version of the paper "An efficient algorithm for the computation of Galois automorphisms", Bill Allombert, Math. Comp, vol. 73, 245, 2001, pp. 359--375.

A group G is said to be "weakly" super-solvable if there exists a normal series

{1} = H_{0} ◃ H_{1} ◃ ... ◃ H_{n-1}
◃ H_{n}

such that each H_{i} is normal in G and for i < n, each quotient group
H_{i+1}/H_{i} is cyclic, and either H_{n} = G (then G is super-solvable) or
G/H_{n} is isomorphic to either A_{4}, S_{4} or the group
(3 x 3):4 (`GAP4(36,9)`

) then
[o_{1},...,o_{g}] ends by [3,3,4].

In practice, almost all small groups are WKSS, the exceptions having order 48(2), 56(1), 60(1), 72(3), 75(1), 80(1), 96(10), 112(1), 120(3) and ≥ 144.

This function is a prerequisite for most of the `galois`

xxx routines.
For instance:

P = x^6 + 108; G = galoisinit(P); L = galoissubgroups(G); vector(#L, i, galoisisabelian(L[i],1)) vector(#L, i, galoisidentify(L[i]))

The output is an 8-component vector *gal*.

*gal*[1] contains the polynomial *pol*
(

).*gal*.pol

*gal*[2] is a three-components vector [p,e,q] where p is a
prime number (

) such that *gal*.p*pol* totally split
modulo p , e is an integer and q = p^e (

) is the
modulus of the roots in *gal*.mod

.*gal*.roots

*gal*[3] is a vector L containing the p-adic roots of
*pol* as integers implicitly modulo

.
(*gal*.mod

).*gal*.roots

*gal*[4] is the inverse of the Vandermonde matrix of the
p-adic roots of *pol*, multiplied by *gal*[5].

*gal*[5] is a multiple of the least common denominator of the
automorphisms expressed as polynomial in a root of *pol*.

*gal*[6] is the Galois group G expressed as a vector of
permutations of L (

).*gal*.group

*gal*[7] is a generating subset S = [s_{1},...,s_{g}] of G
expressed as a vector of permutations of L (

).*gal*.gen

*gal*[8] contains the relative orders [o_{1},...,o_{g}] of
the generators of S (

).*gal*.orders

Let H_{n} be as above, we have the following properties:

***** if G/H_{n} ~ A_{4} then [o_{1},...,o_{g}] ends by
[2,2,3].

***** if G/H_{n} ~ S_{4} then [o_{1},...,o_{g}] ends by
[2,2,3,2].

***** if G/H_{n} ~ (3 x 3):4 (`GAP4(36,9)`

) then
[o_{1},...,o_{g}] ends by [3,3,4].

***** for 1 ≤ i ≤ g the subgroup of G generated by
[s_{1},...,s_{i}] is normal, with the exception of i = g-2 in the
A_{4} case and of i = g-3 in the S_{4} case.

***** the relative order o_{i} of s_{i} is its order in the
quotient group G/`<`

s_{1},...,s_{i-1}`>`

, with the same
exceptions.

***** for any x ∈ G there exists a unique family
[e_{1},...,e_{g}] such that (no exceptions):

-- for 1 ≤ i ≤ g we have 0 ≤ e_{i} < o_{i}

-- x = g_{1}^{e1}g_{2}^{e2}...g_{n}^{en}

If present den must be a suitable value for *gal*[5].

The library syntax is `GEN `

.**galoisinit**(GEN pol, GEN den = NULL)

*gal* being as output by `galoisinit`

, return 0 if
*gal* is not an abelian group, and the HNF matrix of *gal* over
`gal.gen`

if *flag* = 0, 1 if *flag* = 1, and the SNF matrix of *gal*
if *flag* = 2.

This command also accepts subgroups returned by `galoissubgroups`

.

The library syntax is `GEN `

.**galoisisabelian**(GEN gal, long flag)

*gal* being as output by `galoisinit`

, and *subgrp* a subgroup
of *gal* as output by `galoissubgroups`

,return 1 if *subgrp* is a
normal subgroup of *gal*, else return 0.

This command also accepts subgroups returned by `galoissubgroups`

.

The library syntax is `long `

.**galoisisnormal**(GEN gal, GEN subgrp)

*gal* being a
Galois group as output by `galoisinit`

and *perm* a element of
*gal*.group, return the polynomial defining the Galois
automorphism, as output by `nfgaloisconj`

, attached to the
permutation *perm* of the roots *gal*.roots. *perm* can
also be a vector or matrix, in this case, `galoispermtopol`

is
applied to all components recursively.

Note that

G = galoisinit(pol); galoispermtopol(G, G[6])~

is equivalent to `nfgaloisconj(pol)`

, if degree of *pol* is greater
or equal to 2.

The library syntax is `GEN `

.**galoispermtopol**(GEN gal, GEN perm)

Computes the Galois group over Q of the splitting field of
P, that is the smallest field over which P is totally split. P is
assumed to be integral, monic and irreducible; it can
also be given by a nf structure. If d is given, it must be a multiple of
the splitting field degree. The output is compatible with functions expecting
a `galoisinit`

structure.

The library syntax is `GEN `

.**galoissplittinginit**(GEN P, GEN d = NULL)

Computes the subextension L of ℚ(ζ_{n}) fixed by the subgroup
H ⊂ (ℤ/nℤ)^{*}. By the Kronecker-Weber theorem, all abelian number
fields can be generated in this way (uniquely if n is taken to be minimal).
This function output is somewhat canonical, as it returns the minimal
polynomial of a Gaussian period Tr_{ℚ(ζn)/L}(ζ_{n}).

The pair (n, H) is deduced from the parameters (N, H) as follows

***** N an integer: then n = N; H is a generator, i.e. an
integer or an integer modulo n; or a vector of generators.

***** N the output of `znstar`

(n) or `znstar`

(n,1).
H as in the first case above, or a matrix, taken to be a HNF left divisor
of the SNF for (ℤ/nℤ)^{*}
(`N.cyc`

), giving the generators of H in terms of `N.gen`

.

***** N the output of `bnrinit(bnfinit(y), m)`

where m is a
module. H as in the first case, or a matrix taken to be a HNF left
divisor of the SNF for the ray class group modulo m
(of type `N.cyc`

), giving the generators of H in terms of
`N.bid.gen`

( = `N`

.gen if N includes generators).

In this last case, beware that H is understood relatively to N; in
particular, if the infinite place does not divide the module, e.g if m is
an integer, then it is not a subgroup of (ℤ/nℤ)^{*}, but of its quotient by
{± 1}.

If fl = 0, compute a polynomial (in the variable *v*) defining
the subfield of ℚ(ζ_{n}) fixed by the subgroup *H* of (ℤ/nℤ)^{*}.

If fl = 1, compute only the conductor of the abelian extension, as a module.

If fl = 2, output [pol, N], where pol is the polynomial as output when fl = 0 and N the conductor as output when fl = 1.

If fl = 3; output `galoisinit(pol)`

.

The following function can be used to compute all subfields of
ℚ(ζ_{n}) (of exact degree `d`

, if `d`

is set):

subcyclo(n, d = -1)= { my(bnr,L,IndexBound); IndexBound = if (d < 0, n, [d]); bnr = bnrinit(bnfinit(y), [n,[1]]); L = subgrouplist(bnr, IndexBound, 1); vector(#L,i, galoissubcyclo(bnr,L[i])); }

Setting `L = subgrouplist(bnr, IndexBound)`

would produce subfields of
exact conductor n oo .

The library syntax is `GEN `

where **galoissubcyclo**(GEN N, GEN H = NULL, long fl, long v = -1)`v`

is a variable number.

Outputs all the subfields of the Galois group *G*, as a vector.
This works by applying `galoisfixedfield`

to all subgroups. The meaning of
*flag* is the same as for `galoisfixedfield`

.

The library syntax is `GEN `

where **galoissubfields**(GEN G, long flag, long v = -1)`v`

is a variable number.

Outputs all the subgroups of the Galois group `gal`

. A subgroup is a
vector [*gen*, *orders*], with the same meaning
as for *gal*.gen and *gal*.orders. Hence *gen* is a vector of
permutations generating the subgroup, and *orders* is the relatives
orders of the generators. The cardinality of a subgroup is the product of the
relative orders. Such subgroup can be used instead of a Galois group in the
following command: `galoisisabelian`

, `galoissubgroups`

,
`galoisexport`

and `galoisidentify`

.

To get the subfield fixed by a subgroup *sub* of *gal*, use

galoisfixedfield(gal,sub[1])

The library syntax is `GEN `

.**galoissubgroups**(GEN G)

*gc* being the structure returned by `gcharinit`

, return a `t_MAT`

whose columns form a basis of the subgroup of algebraic Grossencharacters in
*gc* (Weil type A0). The last component is interpreted as a power of the
norm.

If *type* is a `t_VEC`

of length *gc*`.r1`

+*gc*`.r2`

,
containing a pair of integers [p_τ,q_τ] for each complex
embedding τ, return a `t_VEC`

containing a character whose infinity type
at τ is

z ` ⟼ `

z^{-p_τ}z^{-q_τ}

if such a character exists, or empty otherwise. The full set of characters of that infinity type is obtained by multiplying by the group of finite order characters.

? bnf = bnfinit(x^4-2*x^3+23*x^2-22*x+6,1); ? gc = gcharinit(bnf,1); ? gc.cyc % = [6, 0, 0, 0, 0.E-57] ? gcharalgebraic(gc) % = [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 0] [0 -1/2 -1/2 -1] ? gcharalgebraic(gc,[[1,1],[0,1]]) % = [] \\ p_τ+q_τ must be constant for an algebraic character to exist ? chi = gcharalgebraic(gc,[[1,1],[0,2]])[1] % = [0, 1, 1, 0, -1]~ ? for(i=0,5,print(lfuneuler([gc,chi+[i,0,0,0,0]~],3))); \\ all characters with this infinity type: multiply by finite order characters

When the torsion subgroup is not cyclic, we can enumerate the characters of a
given type with `forvec`

.

? bnf = bnfinit(x^4+15*x^2+45,1); ? gc = gcharinit(bnf,1); ? gc.cyc % = [2, 2, 0, 0, 0, 0.E-57] ? [chi] = gcharalgebraic(gc,[[2,0],[2,0]]); ? {forvec(v=vectorv(2,i,[0,gc.cyc[i]-1]), print(round(lfunan([gc,chi+concat(v,[0,0,0,0]~)],20))); )}; [1, 0, 0, 4, -5, 0, 0, 0, -9, 0, 16, 0, 0, 0, 0, 16, 0, 0, 16, -20] [1, 0, 0, -4, 5, 0, 0, 0, 9, 0, 16, 0, 0, 0, 0, 16, 0, 0, -16, -20] [1, 0, 0, 4, 5, 0, 0, 0, 9, 0, -16, 0, 0, 0, 0, 16, 0, 0, 16, 20] [1, 0, 0, -4, -5, 0, 0, 0, -9, 0, -16, 0, 0, 0, 0, 16, 0, 0, -16, 20]

Some algebraic Hecke characters are related to CM Abelian varieties. We first show an example with an elliptic curve.

? E = ellinit([0, 0, 1, -270, -1708]); \\ elliptic curve with potential CM by ℚ(sqrt{-3}) ? bnf = bnfinit(x^2+3,1); ? p3 = idealprimedec(bnf,3)[1]; ? gc = gcharinit(bnf,Mat([p3,2])); ? gc.cyc % = [0, 0.E-57] ? [chi] = gcharalgebraic(gc,[[1,0]]) % = [[-1, -1/2]~] ? LE = lfuncreate(E); ? lfunan(LE,20) % = [1, 0, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0, 5, 0, 0, 4, 0, 0, -7, 0] ? Lchi = lfuncreate([gc,chi]); ? round(lfunan(Lchi,20)) % = [1, 0, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0, 5, 0, 0, 4, 0, 0, -7, 0]

Here is an example with a CM Abelian surface.

? L = lfungenus2([-2*x^4 - 2*x^3 + 2*x^2 + 3*x - 2, x^3]); ? bnf = bnfinit(a^4 - a^3 + 2*a^2 + 4*a + 3, 1); ? pr = idealprimedec(bnf,13)[1]; ? gc = gcharinit(bnf,pr); ? gc.cyc % = [3, 0, 0, 0, 0.E-57] ? chitors = [1,0,0,0,0]~; ? typ = [[1,0],[1,0]]; ? [chi0] = gcharalgebraic(gc,typ); ? igood = oo; nbgood = 0; ? {for(i=0,gc.cyc[1]-1, chi = chi0 + i*chitors; Lchi = lfuncreate([gc,chi]); if(lfunparams(L) == lfunparams(Lchi) && exponent(lfunan(L,10) - lfunan(Lchi,10)) < -50, igood=i; nbgood++ ); )}; ? nbgood % = 1 ? chi = chi0 + igood*chitors; ? Lchi = lfuncreate([gc,chi]); ? lfunan(L,30) % = [1, 0, -3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, -3, 0] ? round(lfunan(Lchi,30)) % = [1, 0, -3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, -3, 0]

The library syntax is `GEN `

.**gcharalgebraic**(GEN gc, GEN type = NULL)

Returns the conductor of `chi`

, as a modulus over `gc.bnf`

. This is
the minimum modulus 𝔪 such that
U(𝔪) ⊂ ker(*chi*)
indicating the exact ramification of *chi*.

***** for a real place v, v | 𝔪 iff χ_{v}(-1) = -1.

***** for a finite place 𝔭, the prime power 𝔭^e divides
exactly 𝔪 if e ≥ 0 is the smallest integer such that χ_{𝔭}
(U_{e}) = 1 where U_{0} = ℤ_{𝔭}^ x and U_{i} =
1+𝔭^iℤ_{𝔭} for i > 0.

? bnf = bnfinit(x^2-5,1); ? gc = gcharinit(bnf,[(13*19)^2,[1,1]]); ? gc.cyc % = [8892, 6, 2, 0, 0.E-57] ? chi = [0,0,1,1]~; ? gcharconductor(gc,chi) % = [[61009, 7267; 0, 169], [1, 0]] ? gcharconductor(gc,13*chi) % = [[4693, 559; 0, 13], [1, 0]] ? gcharconductor(gc,13*19*chi) % = [[247, 65; 0, 13], [1, 0]] ? gcharconductor(gc,13*19*168*chi) % = [[19, 5; 0, 1], [0, 0]]

The library syntax is `GEN `

.**gchar_conductor**(GEN gc, GEN chi)

Return internal logarithm vector of character `chi`

as a `t_VEC`

in ℝ^n, so that for all *x*,
`gchareval`

(*gc*,*chi*,*x*,0) is equal to
`gcharduallog`

(*gc*,*chi*) * `gcharlog`

(*gc*,*x*).

The components are organized as follows:

***** the first `ns`

components are in ℝ and describe the character on
the class group generators: θ encodes 𝔭` ⟼ `

exp(2iπθ),

***** the next `nc`

components are in ℝ and describe the `idealstar`

group character via its image on generators: θ encodes the
image exp(2iπθ),

***** the next r_{1}+r_{2} components are in ℝ and correspond to characters
of ℝ for each infinite place: ϕ encodes x` ⟼ `

|x|^{iϕ} in
the real case and z` ⟼ `

|z|^{2iϕ} in the complex case,

***** the last r_{2} components are in ℤ and correspond to characters of
ℝ/ℤ for each complex place: k encodes z` ⟼ `

(z/|z|)^k.

***** the last component s is in ℂ and corresponds to a
power |.|^s of the adélic norm.

See also `gcharlog`

.

? bnf = bnfinit(x^3+4*x-1,1); ? gc = gcharinit(bnf,[1,[1]]); ? gc.cyc % = [2, 0, 0, 0.E-57] ? chi = [0,1,0]~; ? f = gcharduallog(gc,chi) % = [0.153497221319231, 1/2, 0.776369647248353, -0.388184823624176, 1, 0] ? pr = idealprimedec(bnf,2)[1]; ? v = gcharlog(gc,pr); ? exp(2*I*Pi*f*v) % = -0.569867696226731232993110144 - 0.821736459454756074068598760*I ? gchareval(gc,chi,pr) % = -0.569867696226731232993110144 - 0.821736459454756074068598760*I

The library syntax is `GEN `

.**gcharduallog**(GEN gc, GEN chi)

*gc* being the structure returned by `gcharinit`

, *chi* a
character in *gc*, and *x* an ideal of the base field, return the
value χ(x). If *flag* = 1 (default), return a value in ℂ^ x ;
if *flag* = 0, return a value in ℂ/ℤ, normalized so that the real part is
between -1/2 and 1/2.

? bnf = bnfinit(x^2-5); ? gc = gcharinit(bnf,1); ? chi = [1]~; ? pr = idealprimedec(bnf,11)[1]; ? a = gchareval(gc,chi,pr) % = -0.3804107379142448929315340886 - 0.9248176417432464199580504588*I ? b = gchareval(gc,chi,pr,0) % = -0.3121086861831031476247589216 ? a == exp(2*Pi*I*b) %7 = 1

The library syntax is `GEN `

.**gchareval**(GEN gc, GEN chi, GEN x, long flag)

*gc* being a Grossencharacter group as output by `gcharinit`

, Lv
being `t_VEC`

of places v encoded by a `t_INT`

(infinite place) or a prime
ideal structure representing a prime not dividing the modulus of gc (finite
place), and Lchiv being a `t_VEC`

of local characters χ_{v} encoded
by [k,ϕ] with k a `t_INT`

and ϕ a `t_REAL`

or
`t_COMPLEX`

representing x` ⟼ `

sign(x)^k|x|^{iϕ} (real
place) or z` ⟼ `

(z/|z|)^k|z|^{2iϕ}(complex place) or by a `t_REAL`

or `t_COMPLEX`

θ representing 𝔭 ` ⟼ `

exp(2iπ θ)
(finite place), returns a Grossencharacter ψ belonging to g such
that ψ_{v} ~ χ_{v} for all v.
At finite places, in place of a scalar one can provide a `t_VEC`

whose
last component is θ, as output by `gcharlocal`

.
To ensure proper identification, it is recommended to provide all infinite
places together with a set of primes that generate the ray class group of
modulus *gc*`.mod`

.

? bnf = bnfinit(x^2-5,1); ? gc = gcharinit(bnf,1); ? chi = gcharidentify(gc,[2],[[0,13.]]); ? gcharlocal(gc,chi,2) % = [0, 13.057005210545987626926134713745179631] ? pr = idealprimedec(bnf,11)[1]; ? chi = gcharidentify(gc,[pr],[0.3]); ? gchareval(gc,chi,pr,0) % = 0.30000006229129706787363344444425752636

If you know only few digits, it may be a good idea to reduce the current precision to obtain a meaningful result.

? bnf = bnfinit(x^2-5,1); ? gc = gcharinit(bnf,1); ? pr = idealprimedec(bnf,11)[1]; ? chi = gcharidentify(gc,[pr],[0.184760]) % = [-420226]~ \\ unlikely to be meaningful ? gchareval(gc,chi,pr,0) % = 0.18475998070331376194260927294721168954 ? \p 10 realprecision = 19 significant digits (10 digits displayed) ? chi = gcharidentify(gc,[pr],[0.184760]) % = [-7]~ \\ probably what we were looking for ? gchareval(gc,chi,pr,0) % = 0.1847608033 ? \p 38 realprecision = 38 significant digits ? gchareval(gc,chi,pr,0) % = 0.18476080328172203337331245154966763237

The output may be a quasi-character.

? bnf = bnfinit(x^2-2,1); ? gc = gcharinit(bnf,1); gc.cyc % = [0] ? gcharidentify(gc,[1,2],[[0,3.5+1/3*I],[0,-3.5+1/3*I]]) % = [-1, 1/3]~

The library syntax is `GEN `

.**gchar_identify**(GEN gc, GEN Lv, GEN Lchiv, long prec)

*bnf* being a number field output by `bnfinit`

(including
fundamental units), f a modulus, initializes a structure (`gc`

)
describing the group of Hecke Grossencharacters of modulus f.
(As in `idealstar`

, the finite part of the conductor may be given
by a factorization into prime ideals, as produced by `idealfactor`

.)

The following member functions are available
on the result: `.bnf`

is the underlying *bnf*,
`.mod`

the modulus, `.cyc`

its elementary divisors.

The internal representation uses a logarithm map on ideals
ℒ: I → ℝ^n,
so that a Hecke Grossencharacter χ can be described by a n
components vector v via
χ: a ∈ I ` ⟼ `

exp(2iπ v.{ℒ(a)}).

See `gcharlog`

for more details on the map ℒ.

? bnf = bnfinit(polcyclo(5),1); \\ initialize number field ℚ(ζ_{5}) ? pr = idealprimedec(bnf,5)[1]; \\ prime 𝔭 = (1-ζ_{5}) above 5 ? gc = gcharinit(bnf,idealpow(bnf,pr,2)); \\ characters of modulus dividing 𝔭^2 ? gc.cyc \\ structure as an abelian group % = [0,0,0,0.E-57] ? chi = [1,1,-1,0]~; \\ a character ? gcharconductor(gc,chi)[1] % = [5 4 4 4] [0 1 0 0] [0 0 1 0] [0 0 0 1]

Currently, `gc`

is a row vector with 11 components:

*gc*[1] is a matrix whose rows describe a system of generators
of the characters as vectors of ℝ^n, under the above description.

*gc*[2] contains the underlying number field *bnf*
(

).*gc*.bnf

*gc*[3] contains the underlying number field *nf*
(

), possibly stored at higher precision than *gc*.nf*bnf*.

*gc*[4] contains data for computing in (ℤ_{K}/f)^ x .

*gc*[5] is a vector S of prime ideals which generate the class group.

*gc*[6] contains data to compute discrete logarithms with respect to S
in the class group.

*gc*[7] is a vector `[Sunits,m]`

, where `Sunits`

describes
the S-units of *bnf* and m is a relation matrix for internal usage.

*gc*[8] is
`[Vecsmall([evalprec,prec,nfprec]), Vecsmall([ntors,nfree,nalg])]`

caching precisions and various dimensions.

*gc*[9] is a vector describing *gc* as a ℤ-module
via its SNF invariants (

), the last component
representing the norm character.*gc*.cyc

*gc*[10] is a vector `[R,U,Ui]`

allowing to convert characters
from SNF basis to internal combination of generators.

Specifically, a character `chi`

in SNF basis has coordinates
`chi*Ui`

in internal basis (the rows of *gc*[1]).

*gc*[11] = m is the matrix of ℒ(v) for all S-units v.

*gc*[12] = u is an integral base change matrix such that *gc*[1]
corresponds to (mu)^{-1}.

The library syntax is `GEN `

.**gcharinit**(GEN bnf, GEN f, long prec)

*gc* being the structure returned by `gcharinit`

and *chi*
a character on *gc*, return 1 if and only if *chi* is an algebraic
(Weil type A0) character, so that its infinity type at every complex
embedding τ can be written

z `: — >`

z^{-p_τ}z^{-q_τ}

for some pair of integers (p_τ,q_τ).

If *type* is given, set it to the `t_VEC`

of exponents [p_τ,q_τ].

? bnf = bnfinit(x^4+1,1); ? gc = gcharinit(bnf,1); ? gc.cyc % = [0, 0, 0, 0.E-57] ? chi1 = [0,0,1]~; ? gcharisalgebraic(gc,chi1) % = 0 ? gcharlocal(gc,chi1,1) % = [-3, -0.89110698909568455588720672648627467040] ? chi2 = [1,0,0,-3]~; ? gcharisalgebraic(gc,chi2,&typ) % = 1 ? typ % = [[6, 0], [2, 4]] ? gcharlocal(gc,chi2,1) % = [-6, 3*I]

The library syntax is `GEN `

.**gcharisalgebraic**(GEN gc, GEN chi, GEN *type = NULL)

`gc`

being a gchar structure initialised by `gcharinit`

, returns
the local component χ_{v}, where v is either an integer between 1
and r_{1}+r_{2} encoding an infinite place, or a prime ideal structure encoding a
finite place.

***** if v is a real place, χ_{v}(x) = {\rm sign}(x)^k
|x|^{iϕ} is encoded as [k,ϕ];

***** if v is a complex place, χ_{v}(z) = (z/|z|)^k |z|^{2iϕ} is
encoded as [k,ϕ];

***** if v = 𝔭 is a finite place not dividing *gc*`.mod`

,
χ_{v}(π_{v}) = exp(2iπ θ) is encoded as [θ];

***** if v = 𝔭 is a finite place dividing *gc*`.mod`

,
we can define a *bid* structure attached to the multiplicative group
G = (ℤ_{K}/𝔭^k)^{*}, where 𝔭^k divides exactly
*gc*`.mod`

(see `idealstar`

).
Then χ_{v} is encoded as [c_{1},...,c_{n},θ] where [c_{1},...,c_{n}]
defines a character on G
(see `gchareval`

) and χ_{v}(π_{v}) = exp(2iπθ). This *bid*
structure only depends on `gc`

and v (and not on the character χ);
it can be recovered through the optional argument *BID*.

? bnf = bnfinit(x^3-x-1); ? gc = gcharinit(bnf,1); ? gc.cyc % = [0, 0, 0.E-57] ? chi = [0,1,1/3]~; ? pr = idealprimedec(bnf,5)[1]; ? gcharlocal(gc,chi,1) % = [0, -4.8839310048284836274074581373242545693 - 1/3*I] ? gcharlocal(gc,chi,2) % = [6, 2.4419655024142418137037290686621272847 - 1/3*I] ? gcharlocal(gc,chi,pr) % = [0.115465135184293124024408915 + 0.0853833331211293579127218326*I] ? bnf = bnfinit(x^2+1,1); ? pr3 = idealprimedec(bnf,3)[1]; ? pr5 = idealprimedec(bnf,5)[1]; ? gc = gcharinit(bnf,[pr3,2;pr5,3]); ? gc.cyc % = [600, 3, 0, 0.E-57] ? chi = [1,1,1]~; ? gcharlocal(gc,chi,pr3,&bid) % = [1, 1, -21/50] ? bid.cyc % = [24, 3] ? gcharlocal(gc,chi,pr5,&bid) % = [98, -0.30120819117478336291229946188762973702] ? bid.cyc % = [100]

The library syntax is `GEN `

.**gcharlocal**(GEN gc, GEN chi, GEN v, long prec, GEN *BID = NULL)

Return the internal (logarithmic) representation of the ideal x suitable
for computations in gc, as a `t_COL`

in ℝ^n.

Its n = `ns+nc`

+(r_{1}+r_{2})+r_{2}+1 components correspond to a
logarithm map on the group of fractional ideals ℒ: I → ℝ^n, see
`gcharinit`

.

More precisely, let x = (α) ∏ 𝔭_{i}^{ai} a
principalization of x on a set S of primes generating
the class group (see `bnfisprincipal`

),
then the logarithm of x is the `t_COL`

ℒ(x) = [ (a_{i}), log_{f}(α), (log|x/α|_τ)/(2π),
(arg(x/α)_τ)/(2π), (log N(x))/(2π).i ]

where

***** the exponent vector (a_{i}) has `ns`

components, where
`ns`

= #S is the number of prime ideals used to generate the class group,

***** log_{f}(α) is a discrete logarithm of
α in the `idealstar`

group (ℤ_{K}/f)^ x , with `nc`

components,

***** log|x/α|_τ has r_{1}+r_{2} components, one for each
real embedding and pair of complex embeddings τ: K → ℂ
(and |z|_τ = |z|^2 for complex τ).

***** arg{(x/α)_τ} has r_{2} components, one for each
pair of complex embeddings τ: K → ℂ.

***** N(x) is the norm of the ideal x.

? bnf = bnfinit(x^3-x^2+5*x+1,1); ? gc = gcharinit(bnf,3); ? gc.cyc % = [3, 0, 0, 0.E-57] ? chi = [1,1,0,-1]~; ? f = gcharduallog(gc,chi); ? pr = idealprimedec(bnf,5)[1]; ? v = gcharlog(gc,pr) % = [2, -5, -1, 0.0188115475004995312411, -0.0188115475004995312411, -0.840176314833856764413, 0.256149999363388073738*I]~ ? exp(2*I*Pi*f*v) % = -4.5285995080704456583673312 + 2.1193835177957097598574507*I ? gchareval(gc,chi,pr) % = -4.5285995080704456583673312 + 2.1193835177957097598574507*I

The library syntax is `GEN `

.**gcharlog**(GEN gc, GEN x, long prec)

*gc* being a Grossencharacter group output by `gcharinit`

,
recompute its archimedean components ensuring accurate computations to
current precision.

It is advisable to increase the precision before computing several values at large ideals.

The library syntax is `GEN `

.**gcharnewprec**(GEN gc, long prec)

Sum of the two ideals x and y in the number field *nf*. The
result is given in HNF.

? K = nfinit(x^2 + 1); ? a = idealadd(K, 2, x + 1) \\ ideal generated by 2 and 1+I %2 = [2 1] [0 1] ? pr = idealprimedec(K, 5)[1]; \\ a prime ideal above 5 ? idealadd(K, a, pr) \\ coprime, as expected %4 = [1 0] [0 1]

This function cannot be used to add arbitrary ℤ-modules, since it assumes that its arguments are ideals:

? b = Mat([1,0]~); ? idealadd(K, b, b) \\ only square t_MATs represent ideals *** idealadd: nonsquare t_MAT in idealtyp. ? c = [2, 0; 2, 0]; idealadd(K, c, c) \\ nonsense %6 = [2 0] [0 2] ? d = [1, 0; 0, 2]; idealadd(K, d, d) \\ nonsense %7 = [1 0] [0 1]

In the last two examples, we get wrong results since the
matrices c and d do not correspond to an ideal: the ℤ-span of their
columns (as usual interpreted as coordinates with respect to the integer basis
`K.zk`

) is not an O_{K}-module. To add arbitrary ℤ-modules generated
by the columns of matrices A and B, use `mathnf(concat(A,B))`

.

The library syntax is `GEN `

.**idealadd**(GEN nf, GEN x, GEN y)

x and y being two co-prime integral ideals (given in any form), this gives a two-component row vector [a,b] such that a ∈ x, b ∈ y and a+b = 1.

The alternative syntax `idealaddtoone`

(*nf*,v), is supported, where
v is a k-component vector of ideals (given in any form) which sum to
ℤ_{K}. This outputs a k-component vector e such that e[i] ∈ x[i] for
1 ≤ i ≤ k and ∑_{1 ≤ i ≤ k}e[i] = 1.

The library syntax is `GEN `

.**idealaddtoone0**(GEN nf, GEN x, GEN y = NULL)

If x is a fractional ideal
(given in any form), gives an element α in *nf* such that for
all prime ideals 𝔭 such that the valuation of x at 𝔭 is
nonzero, we have v_{𝔭}(α) = v_{𝔭}(x), and
v_{𝔭}(α) ≥ 0 for all other 𝔭.

The argument x may also be given as a prime ideal factorization, as
output by `idealfactor`

, but allowing zero exponents.
This yields an element α such that for all prime ideals 𝔭
occurring in x, v_{𝔭}(α) = v_{𝔭}(x);
for all other prime ideals, v_{𝔭}(α) ≥ 0.

flag is deprecated (ignored), kept for backward compatibility.

The library syntax is `GEN `

.
Use directly **idealappr0**(GEN nf, GEN x, long flag)`GEN `

since **idealappr**(GEN nf, GEN x)*flag* is ignored.

x being a prime ideal factorization (i.e. a 2-columns matrix whose first
column contains prime ideals and the second column contains integral
exponents), y a vector of elements in *nf* indexed by the ideals in
x, computes an element b such that

v_{𝔭}(b - y_{𝔭}) ≥ v_{𝔭}(x) for all prime ideals
in x and v_{𝔭}(b) ≥ 0 for all other 𝔭.

? K = nfinit(t^2-2); ? x = idealfactor(K, 2^2*3) %2 = [[2, [0, 1]~, 2, 1, [0, 2; 1, 0]] 4] [ [3, [3, 0]~, 1, 2, 1] 1] ? y = [t,1]; ? idealchinese(K, x, y) %4 = [4, -3]~

The argument x may also be of the form [x, s] where the first component
is as above and s is a vector of signs, with r_{1} components
s_{i} in {-1,0,1}:
if σ_{i} denotes the i-th real embedding of the number field,
the element b returned satisfies further
`sign`

(σ_{i}(b)) = s_{i} for all i such that s_{i} = ±1.
In other words, the sign is fixed to s_{i} at the i-th embedding whenever
s_{i} is nonzero.

? idealchinese(K, [x, [1,1]], y) %5 = [16, -3]~ ? idealchinese(K, [x, [-1,-1]], y) %6 = [-20, -3]~ ? idealchinese(K, [x, [1,-1]], y) %7 = [4, -3]~

If y is omitted, return a data structure which can be used in place of x in later calls and allows to solve many chinese remainder problems for a given x more efficiently. In this case, the right hand side y is not allowed to have denominators, unless they are coprime to x.

? C = idealchinese(K, [x, [1,1]]); ? idealchinese(K, C, y) \\ as above %9 = [16, -3]~ ? for(i=1,10^4, idealchinese(K,C,y)) \\ ... but faster ! time = 80 ms. ? for(i=1,10^4, idealchinese(K,[x,[1,1]],y)) time = 224 ms.

Finally, this structure is itself allowed in place of x, the new s overriding the one already present in the structure. This allows to initialize for different sign conditions more efficiently when the underlying ideal factorization remains the same.

? D = idealchinese(K, [C, [1,-1]]); \\ replaces [1,1] ? idealchinese(K, D, y) %13 = [4, -3]~ ? for(i=1,10^4,idealchinese(K,[C,[1,-1]])) time = 40 ms. \\ faster than starting from scratch ? for(i=1,10^4,idealchinese(K,[x,[1,-1]])) time = 128 ms.

The library syntax is `GEN `

.
Also available is
**idealchinese**(GEN nf, GEN x, GEN y = NULL)`GEN `

when y = **idealchineseinit**(GEN nf, GEN x)`NULL`

.

Given two integral ideals x and y
in the number field *nf*, returns a β in the field,
such that β.x is an integral ideal coprime to y. In fact,
β is also guaranteed to be integral outside primes dividing y.

The library syntax is `GEN `

.**idealcoprime**(GEN nf, GEN x, GEN y)

Quotient x.y^{-1} of the two ideals x and y in the number
field *nf*. The result is given in HNF.

If *flag* is nonzero, the quotient x.y^{-1} is assumed to be an
integral ideal. This can be much faster when the norm of the quotient is
small even though the norms of x and y are large. More precisely,
the algorithm cheaply removes all maximal ideals above rational
primes such that v_{p}(Nx) = v_{p}(Ny).

The library syntax is `GEN `

.
Also available are **idealdiv0**(GEN nf, GEN x, GEN y, long flag)`GEN `

(**idealdiv**(GEN nf, GEN x, GEN y)*flag* = 0) and `GEN `

(**idealdivexact**(GEN nf, GEN x, GEN y)*flag* = 1).

Let *nf* be a number field as output by `nfinit`

, and x a
fractional ideal. This function returns the nonnegative rational generator
of x ∩ ℚ. If x is an extended ideal, the extended part is ignored.

? nf = nfinit(y^2+1); ? idealdown(nf, -1/2) %2 = 1/2 ? idealdown(nf, (y+1)/3) %3 = 2/3 ? idealdown(nf, [2, 11]~) %4 = 125 ? x = idealprimedec(nf, 2)[1]; idealdown(nf, x) %5 = 2 ? idealdown(nf, [130, 94; 0, 2]) %6 = 130

The library syntax is `GEN `

.**idealdown**(GEN nf, GEN x)

Factors into prime ideal powers the ideal x in the number field
*nf*. The output format is similar to the `factor`

function, and
the prime ideals are represented in the form output by the
`idealprimedec`

function. If *lim* is set, return partial
factorization, including only prime ideals above rational primes
< *lim*.

? nf = nfinit(x^3-2); ? idealfactor(nf, x) \\ a prime ideal above 2 %2 = [[2, [0, 1, 0]~, 3, 1, ...] 1] ? A = idealhnf(nf, 6*x, 4+2*x+x^2) %3 = [6 0 4] [0 6 2] [0 0 1] ? idealfactor(nf, A) %4 = [[2, [0, 1, 0]~, 3, 1, ...] 2] [[3, [1, 1, 0]~, 3, 1, ...] 2] ? idealfactor(nf, A, 3) \\ restrict to primes above p < 3 %5 = [[2, [0, 1, 0]~, 3, 1, ...] 2]

The library syntax is `GEN `

.
This function should only be used by the **gpidealfactor**(GEN nf, GEN x, GEN lim = NULL)`gp`

interface. Use
directly `GEN `

or
**idealfactor**(GEN nf, GEN x)`GEN `

.**idealfactor_limit**(GEN nf, GEN x, ulong lim)

Gives back the ideal corresponding to a factorization. The integer 1
corresponds to the empty factorization.
If e is present, e and f must be vectors of the same length (e being
integral), and the corresponding factorization is the product of the
f[i]^{e[i]}.

If not, and f is vector, it is understood as in the preceding case with e
a vector of 1s: we return the product of the f[i]. Finally, f can be a
regular factorization, as produced by `idealfactor`

.

? nf = nfinit(y^2+1); idealfactor(nf, 4 + 2*y) %1 = [[2, [1, 1]~, 2, 1, [1, 1]~] 2] [[5, [2, 1]~, 1, 1, [-2, 1]~] 1] ? idealfactorback(nf, %) %2 = [10 4] [0 2] ? f = %1[,1]; e = %1[,2]; idealfactorback(nf, f, e) %3 = [10 4] [0 2] ? % == idealhnf(nf, 4 + 2*y) %4 = 1

If `flag`

is nonzero, perform ideal reductions (`idealred`

) along the
way. This is most useful if the ideals involved are all *extended*
ideals (for instance with trivial principal part), so that the principal parts
extracted by `idealred`

are not lost. Here is an example:

? f = vector(#f, i, [f[i], [;]]); \\ transform to extended ideals ? idealfactorback(nf, f, e, 1) %6 = [[1, 0; 0, 1], [2, 1; [2, 1]~, 1]] ? nffactorback(nf, %[2]) %7 = [4, 2]~

The extended ideal returned in `%6`

is the trivial ideal 1, extended
with a principal generator given in factored form. We use `nffactorback`

to recover it in standard form.

The library syntax is `GEN `

.**idealfactorback**(GEN nf, GEN f, GEN e = NULL, long flag)

Let K be the number field defined by nf and assume K/ℚ be a
Galois extension with Galois group given `gal = galoisinit(nf)`

,
and that *pr* is an unramified prime ideal 𝔭 in `prid`

format.
This function returns a permutation of `gal.group`

which defines
the Frobenius element Frob_{𝔭} attached to 𝔭.
If p is the unique prime number in 𝔭, then
Frob(x) = x^p mod 𝔭 for all x ∈ ℤ_{K}.

? nf = nfinit(polcyclo(31)); ? gal = galoisinit(nf); ? pr = idealprimedec(nf,101)[1]; ? g = idealfrobenius(nf,gal,pr); ? galoispermtopol(gal,g) %5 = x^8

This is correct since 101 = 8 mod 31.

The library syntax is `GEN `

.**idealfrobenius**(GEN nf, GEN gal, GEN pr)

Gives the Hermite normal form of the ideal uℤ_{K}+vℤ_{K}, where u
and v are elements of the number field K defined by *nf*.

? nf = nfinit(y^3 - 2); ? idealhnf(nf, 2, y+1) %2 = [1 0 0] [0 1 0] [0 0 1] ? idealhnf(nf, y/2, [0,0,1/3]~) %3 = [1/3 0 0] [0 1/6 0] [0 0 1/6]

If b is omitted, returns the HNF of the ideal defined by u: u may be an
algebraic number (defining a principal ideal), a maximal ideal (as given by
`idealprimedec`

or `idealfactor`

), or a matrix whose columns give
generators for the ideal. This last format is a little complicated, but
useful to reduce general modules to the canonical form once in a while:

***** if strictly less than N = [K:ℚ] generators are given, u
is the ℤ_{K}-module they generate,

***** if N or more are given, it is *assumed* that they form a
ℤ-basis of the ideal, in particular that the matrix has maximal rank N.
This acts as `mathnf`

since the ℤ_{K}-module structure is (taken for
granted hence) not taken into account in this case.

? idealhnf(nf, idealprimedec(nf,2)[1]) %4 = [2 0 0] [0 1 0] [0 0 1] ? idealhnf(nf, [1,2;2,3;3,4]) %5 = [1 0 0] [0 1 0] [0 0 1]

Finally, when K is quadratic with discriminant D_{K}, we
allow u = `Qfb(a,b,c)`

, provided b^2 - 4ac = D_{K}. As usual,
this represents the ideal a ℤ + (1/2)(-b + sqrt{D_{K}}) ℤ.

? K = nfinit(x^2 - 60); K.disc %1 = 60 ? idealhnf(K, qfbprimeform(60,2)) %2 = [2 1] [0 1] ? idealhnf(K, Qfb(1,2,3)) *** at top-level: idealhnf(K,Qfb(1,2,3 *** ^ — — — — — — -- *** idealhnf: Qfb(1, 2, 3) has discriminant != 60 in idealhnf.

The library syntax is `GEN `

.
Also available is **idealhnf0**(GEN nf, GEN u, GEN v = NULL)`GEN `

, where **idealhnf**(GEN nf, GEN a)`nf`

is a true *nf* structure.

Intersection of the two ideals
A and B in the number field *nf*. The result is given in HNF.

? nf = nfinit(x^2+1); ? idealintersect(nf, 2, x+1) %2 = [2 0] [0 2]

This function does not apply to general ℤ-modules, e.g. orders, since its arguments are replaced by the ideals they generate. The following script intersects ℤ-modules A and B given by matrices of compatible dimensions with integer coefficients:

ZM_intersect(A,B) = { my(Ker = matkerint(concat(A,B))); mathnf( A * Ker[1..#A,] ) }

The library syntax is `GEN `

.**idealintersect**(GEN nf, GEN A, GEN B)

Inverse of the ideal x in the
number field *nf*, given in HNF. If x is an extended
ideal, its principal part is suitably
updated: i.e. inverting [I,t], yields [I^{-1}, 1/t].

The library syntax is `GEN `

.**idealinv**(GEN nf, GEN x)

Given *nf* a number field as output by `nfinit`

and an ideal
x, return 0 if x is not a maximal ideal. Otherwise return a `prid`

structure *nf* attached to the ideal. This function uses
`ispseudoprime`

and may return a wrong result in case the underlying
rational pseudoprime is not an actual prime number: apply `isprime(pr.p)`

to guarantee correctness. If x is an extended ideal, the extended part is
ignored.

? K = nfinit(y^2 + 1); ? idealismaximal(K, 3) \\ 3 is inert %2 = [3, [3, 0]~, 1, 2, 1] ? idealismaximal(K, 5) \\ 5 is not %3 = 0 ? pr = idealprimedec(K,5)[1] \\ already a prid %4 = [5, [-2, 1]~, 1, 1, [2, -1; 1, 2]] ? idealismaximal(K, pr) \\ trivial check %5 = [5, [-2, 1]~, 1, 1, [2, -1; 1, 2]] ? x = idealhnf(K, pr) %6 = [5 3] [0 1] ? idealismaximal(K, x) \\ converts from matrix form to prid %7 = [5, [-2, 1]~, 1, 1, [2, -1; 1, 2]]

This function is noticeably faster than `idealfactor`

since it never involves an actually factorization, in particular when x
∩ ℤ is not a prime number.

The library syntax is `GEN `

.**idealismaximal**(GEN nf, GEN x)

Let *nf* be a number field and n > 0 be a positive integer.
Return 1 if the fractional ideal A = B^n is an n-th power and 0
otherwise. If the argument B is present, set it to the n-th root of A,
in HNF.

? K = nfinit(x^3 - 2); ? A = [46875, 30966, 9573; 0, 3, 0; 0, 0, 3]; ? idealispower(K, A, 3, &B) %3 = 1 ? B %4 = [75 22 41] [ 0 1 0] [ 0 0 1] ? A = [9375, 2841, 198; 0, 3, 0; 0, 0, 3]; ? idealispower(K, A, 3) %5 = 0

The library syntax is `long `

.**idealispower**(GEN nf, GEN A, long n, GEN *B = NULL)

Computes the list of all ideals of norm less or equal to *bound* in
the number field
*nf*. The result is a row vector with exactly *bound* components.
Each component is itself a row vector containing the information about
ideals of a given norm, in no specific order. The information is inferred
from local data and Chinese remainders and less expensive than computing
than a direct global computation.

The binary digits of *flag* mean:

***** 1: if the ideals are given by a *bid*, include generators;
otherwise don't.

***** 2: if this bit is set, *nf* must be a *bnf* with units. Each
component is of the form [*bid*,U], where *bid* is attached to
an ideal f and U is a vector of discrete logarithms of the units in
(ℤ_{K}/f)^{*}. More precisely, U gives the `ideallog`

s with respect
to *bid* of (ζ,u_{1},...,u_{r})
where ζ is the torsion unit generator `bnf.tu[2]`

and (u_{i})
are the fundamental units in `bnf.fu`

.
This structure is technical, meant to be used in conjunction with
`bnrclassnolist`

or `bnrdisclist`

.

***** 4: give only the ideal (in HNF), else a *bid*.

***** 8: omit ideals which cannot be conductors, i.e. divisible exactly my
a prime ideal of norm 2.

? nf = nfinit(x^2+1); ? L = ideallist(nf, 100); ? L[1] %3 = [[1, 0; 0, 1]] \\ A single ideal of norm 1 ? #L[65] %4 = 4 \\ There are 4 ideals of norm 65 in ℤ[i]

If one wants more information:

? L = ideallist(nf, 100, 0); ? l = L[25]; vector(#l, i, l[i].clgp) %6 = [[20, [20]], [16, [4, 4]], [20, [20]]] ? l[1].mod %7 = [[25, 18; 0, 1], []] ? l[2].mod %8 = [[5, 0; 0, 5], []] ? l[3].mod %9 = [[25, 7; 0, 1], []]

where we ask for the structures of the (ℤ[i]/f)^{*} for all
three ideals of norm 25. In fact, for all moduli with finite part of norm
25 and trivial Archimedean part, as the last 3 commands show. See
`ideallistarch`

to treat general moduli.

Finally, on can input a negative `bound`

. The function
then returns the ideals of norm |`bound`

|, given by their
factorization matrix. If needed, one can obtain their HNF using
`idealfactorback`

, and the corresponding *bid* structures using
`idealstar`

(which accepts ideals in factored form).

The library syntax is `GEN `

.
Also available is
**gideallist**(GEN nf, GEN bound, long flag)`GEN `

for a non-negative
bound.**ideallist0**(GEN nf,long bound, long flag)

*list* is a vector of vectors of bid's, as output by `ideallist`

with
flag 0 to 3. Return a vector of vectors with the same number of
components as the original *list*. The leaves give information about
moduli whose finite part is as in original list, in the same order, and
Archimedean part is now *arch* (it was originally trivial). The
information contained is of the same kind as was present in the input; see
`ideallist`

, in particular the meaning of *flag*.

? bnf = bnfinit(x^2-2); ? bnf.sign %2 = [2, 0] \\ two places at infinity ? L = ideallist(bnf, 100, 0); ? l = L[98]; vector(#l, i, l[i].clgp) %4 = [[42, [42]], [36, [6, 6]], [42, [42]]] ? La = ideallistarch(bnf, L, [1,1]); \\ add them to the modulus ? l = La[98]; vector(#l, i, l[i].clgp) %6 = [[168, [42, 2, 2]], [144, [6, 6, 2, 2]], [168, [42, 2, 2]]]

Of course, the results above are obvious: adding t places at infinity will
add t copies of ℤ/2ℤ to (ℤ_{K}/f)^{*}. The following application
is more typical:

? L = ideallist(bnf, 100, 2); \\ units are required now ? La = ideallistarch(bnf, L, [1,1]); ? H = bnrclassnolist(bnf, La); ? H[98]; %4 = [2, 12, 2]

The library syntax is `GEN `

.**ideallistarch**(GEN nf, GEN list, GEN arch)

*nf* is a number field,
*bid* is as output by `idealstar(nf, D,...)`

and x an
element of *nf* which must have valuation
equal to 0 at all prime ideals in the support of `D`

and need not be
integral. This function
computes the discrete logarithm of x on the generators given in

. In other words, if g*bid*.gen_{i} are these generators, of orders
d_{i} respectively, the result is a column vector of integers (x_{i}) such
that 0 ≤ x_{i} < d_{i} and
x = ∏_{i} g_{i}^{xi} (mod ^{*}D) .
Note that when the support of `D`

contains places at infinity, this
congruence implies also sign conditions on the attached real embeddings.
See `znlog`

for the limitations of the underlying discrete log algorithms.

When *nf* is omitted, take it to be the rational number field. In that
case, x must be a `t_INT`

and *bid* must have been initialized by
`znstar(N,1)`

.

The library syntax is `GEN `

.
Also available are
**ideallog**(GEN nf = NULL, GEN x, GEN bid)`GEN `

when **Zideallog**(GEN bid, GEN x)`nf`

is `NULL`

,
and `GEN `

that returns the discrete logarithm of x modulo the **ideallogmod**(GEN nf, GEN x, GEN bid, GEN mod)`t_INT`

`mod`

; the value `mod = NULL`

is treated as 0 (full discrete
logarithm), but `nf = NULL`

is not implemented with nonzero `mod`

.

*This function is useless and kept for backward compatibility only,
use idealred*. Computes a pseudo-minimum of the ideal x in the
direction

The library syntax is `GEN `

.**idealmin**(GEN nf, GEN ix, GEN vdir = NULL)

Ideal multiplication of the ideals x and y in the number field
*nf*; the result is the ideal product in HNF. If either x or y
are extended ideals, their principal part is suitably
updated: i.e. multiplying [I,t], [J,u] yields [IJ, tu]; multiplying
I and [J, u] yields [IJ, u].

? nf = nfinit(x^2 + 1); ? idealmul(nf, 2, x+1) %2 = [4 2] [0 2] ? idealmul(nf, [2, x], x+1) \\ extended ideal * ideal %3 = [[4, 2; 0, 2], x] ? idealmul(nf, [2, x], [x+1, x]) \\ two extended ideals %4 = [[4, 2; 0, 2], [-1, 0]~]

If *flag* is nonzero, reduce the result using `idealred`

.

The library syntax is `GEN `

.**idealmul0**(GEN nf, GEN x, GEN y, long flag)

See also
`GEN `

(**idealmul**(GEN nf, GEN x, GEN y)*flag* = 0) and
`GEN `

(**idealmulred**(GEN nf, GEN x, GEN y)*flag* != 0).

Computes the norm of the ideal x in the number field *nf*.

The library syntax is `GEN `

.**idealnorm**(GEN nf, GEN x)

Returns [A,B], where A,B are coprime integer ideals
such that x = A/B, in the number field *nf*.

? nf = nfinit(x^2+1); ? idealnumden(nf, (x+1)/2) %2 = [[1, 0; 0, 1], [2, 1; 0, 1]]

The library syntax is `GEN `

.**idealnumden**(GEN nf, GEN x)

Computes the k-th power of
the ideal x in the number field *nf*; k ∈ ℤ.
If x is an extended
ideal, its principal part is suitably
updated: i.e. raising [I,t] to the k-th power, yields [I^k, t^k].

If *flag* is nonzero, reduce the result using `idealred`

, *throughout
the (binary) powering process*; in particular, this is *not* the same
as `idealpow`

(*nf*,x,k) followed by reduction.

The library syntax is `GEN `

.**idealpow0**(GEN nf, GEN x, GEN k, long flag)

See also
`GEN `

and
**idealpow**(GEN nf, GEN x, GEN k)`GEN `

(**idealpows**(GEN nf, GEN x, long k)*flag* = 0).
Corresponding to *flag* = 1 is `GEN `

.**idealpowred**(GEN nf, GEN vp, GEN k)

Computes the prime ideal
decomposition of the (positive) prime number p in the number field K
represented by *nf*. If a nonprime p is given the result is undefined.
If f is present and nonzero, restrict the result to primes of residue
degree ≤ f.

The result is a vector of *prid* structures, each representing one of the
prime ideals above p in the number field *nf*. The representation
`pr`

= [p,a,e,f,*mb*] of a prime ideal means the following: a
is an algebraic integer in the maximal order ℤ_{K} and the prime ideal is
equal to 𝔭 = pℤ_{K} + aℤ_{K};
e is the ramification index; f is the residual index;
finally, *mb* is the multiplication table attached to an algebraic
integer b such that 𝔭^{-1} = ℤ_{K}+ b/ pℤ_{K}, which is used
internally to compute valuations. In other words if p is inert,
then *mb* is the integer 1, and otherwise it is a square `t_MAT`

whose j-th column is b.`nf.zk[j]`

.

The algebraic number a is guaranteed to have a valuation equal to 1 at the prime ideal (this is automatic if e > 1).

The components of `pr`

should be accessed by member functions: `pr.p`

,
`pr.e`

, `pr.f`

, and `pr.gen`

(returns the vector [p,a]):

? K = nfinit(x^3-2); ? P = idealprimedec(K, 5); ? #P \\ 2 primes above 5 in Q(2^(1/3)) %3 = 2 ? [p1,p2] = P; ? [p1.e, p1.f] \\ the first is unramified of degree 1 %5 = [1, 1] ? [p2.e, p2.f] \\ the second is unramified of degree 2 %6 = [1, 2] ? p1.gen %7 = [5, [2, 1, 0]~] ? nfbasistoalg(K, %[2]) \\ a uniformizer for p1 %8 = Mod(x + 2, x^3 - 2) ? #idealprimedec(K, 5, 1) \\ restrict to f = 1 %9 = 1 \\ now only p1

The library syntax is `GEN `

.**idealprimedec_limit _{f}**(GEN nf, GEN p, long f)

Given a prime ideal in `idealprimedec`

format,
returns the multiplicative group (1 + *pr*) / (1 + *pr*^k) as an
abelian group. This function is much faster than `idealstar`

when the
norm of *pr* is large, since it avoids (useless) work in the
multiplicative group of the residue field.

? K = nfinit(y^2+1); ? P = idealprimedec(K,2)[1]; ? G = idealprincipalunits(K, P, 20); ? G.cyc %4 = [512, 256, 4] \\ Z/512 x Z/256 x Z/4 ? G.gen %5 = [[-1, -2]~, 1021, [0, -1]~] \\ minimal generators of given order

The library syntax is `GEN `

.**idealprincipalunits**(GEN nf, GEN pr, long k)

Let K be the number field defined by *nf* and assume that K/ℚ is
Galois with Galois group G given by `gal = galoisinit(nf)`

.
Let *pr* be the prime ideal 𝔓 in prid format.
This function returns a vector g of subgroups of `gal`

as follows:

***** `g[1]`

is the decomposition group of 𝔓,

***** `g[2]`

is G_{0}(𝔓), the inertia group of 𝔓,

and for i ≥ 2,

***** `g[i]`

is G_{i-2}(𝔓), the i-2-th
ramification group of 𝔓.

The length of g is the number of nontrivial groups in the sequence, thus is 0 if e = 1 and f = 1, and 1 if f > 1 and e = 1. The following function computes the cardinality of a subgroup of G, as given by the components of g:

card(H) =my(o=H[2]); prod(i=1,#o,o[i]);

? nf=nfinit(x^6+3); gal=galoisinit(nf); pr=idealprimedec(nf,3)[1]; ? g = idealramgroups(nf, gal, pr); ? apply(card,g) %3 = [6, 6, 3, 3, 3] \\ cardinalities of the G_{i}

? nf=nfinit(x^6+108); gal=galoisinit(nf); pr=idealprimedec(nf,2)[1]; ? iso=idealramgroups(nf,gal,pr)[2] %5 = [[Vecsmall([2, 3, 1, 5, 6, 4])], Vecsmall([3])] ? nfdisc(galoisfixedfield(gal,iso,1)) %6 = -3

The field fixed by the inertia group of 2 is not ramified at 2.

The library syntax is `GEN `

.**idealramgroups**(GEN nf, GEN gal, GEN pr)

LLL reduction of
the ideal I in the number field K attached to *nf*, along the
direction v. The v parameter is best left omitted, but if it is present,
it must be an `nf.r1`

+ `nf.r2`

-component vector of
*nonnegative* integers. (What counts is the relative magnitude of the
entries: if all entries are equal, the effect is the same as if the vector
had been omitted.)

This function finds an a ∈ K^{*} such that J = (a)I is
"small" and integral (see the end for technical details).
The result is the Hermite normal form of
the "reduced" ideal J.

? K = nfinit(y^2+1); ? P = idealprimedec(K,5)[1]; ? idealred(K, P) %3 = [1 0] [0 1]

More often than not, a principal ideal yields the unit
ideal as above. This is a quick and dirty way to check if ideals are principal,
but it is not a necessary condition: a nontrivial result does not prove that
the ideal is nonprincipal. For guaranteed results, see `bnfisprincipal`

,
which requires the computation of a full `bnf`

structure.

If the input is an extended ideal [I,s], the output is [J, sa]; in this way, one keeps track of the principal ideal part:

? idealred(K, [P, 1]) %5 = [[1, 0; 0, 1], [2, -1]~]

meaning that P is generated by [2, -1] . The number field element in the
extended part is an algebraic number in any form *or* a factorization
matrix (in terms of number field elements, not ideals!). In the latter case,
elements stay in factored form, which is a convenient way to avoid
coefficient explosion; see also `idealpow`

.

**Technical note.** The routine computes an LLL-reduced
basis for the lattice I^{-1} equipped with the quadratic
form
|| x ||_{v}^2 = ∑_{i = 1}^{r1+r2} 2^{vi}ϵ_{i}|σ_{i}(x)|^2,
where as usual the σ_{i} are the (real and) complex embeddings and
ϵ_{i} = 1, resp. 2, for a real, resp. complex place. The element
a is simply the first vector in the LLL basis. The only reason you may want
to try to change some directions and set some v_{i} != 0 is to randomize
the elements found for a fixed ideal, which is heuristically useful in index
calculus algorithms like `bnfinit`

and `bnfisprincipal`

.

**Even more technical note.** In fact, the above is a white lie.
We do not use ||.||_{v} exactly but a rescaled rounded variant which
gets us faster and simpler LLLs. There's no harm since we are not using any
theoretical property of a after all, except that it belongs to I^{-1}
and that a I is "expected to be small".

The library syntax is `GEN `

.**idealred0**(GEN nf, GEN I, GEN v = NULL)

Let *nf* be a number field, x an ideal in *nf* and n > 0 be a
positive integer. Return a number field element b such that x b^n = v
is small. If x is integral, then v is also integral.

More precisely, `idealnumden`

reduces the problem to x integral. Then,
factoring out the prime ideals dividing a rational prime p ≤ B,
we rewrite x = I J^n where the ideals I and J are both integral and
I is B-smooth. Then we return a small element b in J^{-1}.

The bound B avoids a costly complete factorization of x; as soon as the n-core of x is B-smooth (i.e., as soon as I is n-power free), then J is as large as possible and so is the expected reduction.

? T = x^6+108; nf = nfinit(T); a = Mod(x,T); ? setrand(1); u = (2*a^2+a+3)*random(2^1000*x^6)^6; ? sizebyte(u) %3 = 4864 ? b = idealredmodpower(nf,u,2); ? v2 = nfeltmul(nf,u, nfeltpow(nf,b,2)) %5 = [34, 47, 15, 35, 9, 3]~ ? b = idealredmodpower(nf,u,6); ? v6 = nfeltmul(nf,u, nfeltpow(nf,b,6)) %7 = [3, 0, 2, 6, -7, 1]~

The last element `v6`

, obtained by reducing
modulo 6-th powers instead of squares, looks smaller than `v2`

but its norm is actually a little larger:

? idealnorm(nf,v2) %8 = 81309 ? idealnorm(nf,v6) %9 = 731781

The library syntax is `GEN `

.**idealredmodpower**(GEN nf, GEN x, ulong n, ulong B)

Outputs a `bid`

structure,
necessary for computing in the finite abelian group G = (ℤ_{K}/N)^{*}. Here,
*nf* is a number field and N is a *modulus*: either an ideal in any
form, or a row vector whose first component is an ideal and whose second
component is a row vector of r_{1} 0 or 1. Ideals can also be given
by a factorization into prime ideals, as produced by `idealfactor`

.

If the positive integer `cycmod`

is present, only compute the group
modulo `cycmod`

-th powers, which may save a lot of time when some
maximal ideals in the modulus have a huge residue field. Whereas you might
only be interested in quadratic or cubic residuosity; see also `bnrinit`

for applications in class field theory.

This *bid* is used in `ideallog`

to compute discrete logarithms. It
also contains useful information which can be conveniently retrieved as

(the modulus),
*bid*.mod

(G as a finite abelian group),
*bid*.clgp

(the cardinality of G),
*bid*.no

(elementary divisors) and
*bid*.cyc

(generators).*bid*.gen

If *flag* = 1 (default), the result is a `bid`

structure without
generators: they are well defined but not explicitly computed, which saves
time.

If *flag* = 2, as *flag* = 1, but including generators.

If *flag* = 0, only outputs (ℤ_{K}/N)^{*} as an abelian group,
i.e as a 3-component vector [h,d,g]: h is the order, d is the vector of
SNF cyclic components and g the corresponding
generators.

If *nf* is omitted, we take it to be the rational number fields, N must
be an integer and we return the structure of (ℤ/Nℤ)^{*}. In other words
`idealstar(, N, flag)`

is short for

idealstar(nfinit(x), N, flag)

but faster. The alternative syntax `znstar(N, flag)`

is also available for an analogous effect but, due to an unfortunate
historical oversight, the default value of `flag`

is different in
the two functions (`znstar`

does not initialize by default, you probably
want `znstar(N,1)`

).

The library syntax is `GEN `

.
Instead the above hardcoded numerical flags, one should rather use
**idealstarmod**(GEN nf = NULL, GEN N, long flag, GEN cycmod = NULL)`GEN `

or
**Idealstarmod**(GEN nf, GEN ideal, long flag, GEN cycmod)`GEN `

(**Idealstar**(GEN nf, GEN ideal, long flag)`cycmod`

is
`NULL`

), where `flag`

is
an or-ed combination of `nf_GEN`

(include generators) and `nf_INIT`

(return a full `bid`

, not a group), possibly 0. This offers
one more combination: gen, but no init. The `nf`

argument must be a true
*nf* structure.

Computes a two-element representation of the ideal x in the number
field *nf*, combining a random search and an approximation theorem; x
is an ideal in any form (possibly an extended ideal, whose principal part is
ignored)

***** When called as `idealtwoelt(nf,x)`

, the result is a row vector
[a,α] with two components such that x = aℤ_{K}+αℤ_{K} and a is
chosen to be the positive generator of x∩ℤ, unless x was given as a
principal ideal in which case we may choose a = 0. The algorithm
uses a fast lazy factorization of x∩ ℤ and runs in randomized
polynomial time.

? K = nfinit(t^5-23); ? x = idealhnf(K, t^2*(t+1), t^3*(t+1)) %2 = \\ some random ideal of norm 552*23 [552 23 23 529 23] [ 0 23 0 0 0] [ 0 0 1 0 0] [ 0 0 0 1 0] [ 0 0 0 0 1] ? [a,alpha] = idealtwoelt(K, x) %3 = [552, [23, 0, 1, 0, 0]~] ? nfbasistoalg(K, alpha) %4 = Mod(t^2 + 23, t^5 - 23)

***** When called as `idealtwoelt(nf,x,a)`

with an explicit nonzero a
supplied as third argument, the function assumes that a ∈ x and returns
α ∈ x such that x = aℤ_{K} + αℤ_{K}. Note that we must factor
a in this case, and the algorithm is generally slower than the
default variant and gives larger generators:

? alpha2 = idealtwoelt(K, x, 552) %5 = [-161, -161, -183, -207, 0]~ ? idealhnf(K, 552, alpha2) == x %6 = 1

Note that, in both cases, the return value is *not*
recognized as an ideal by GP functions; one must use `idealhnf`

as
above to recover a valid ideal structure from the two-element representation.

The library syntax is `GEN `

.
Also available are
**idealtwoelt0**(GEN nf, GEN x, GEN a = NULL)`GEN `

and
**idealtwoelt**(GEN nf, GEN x)`GEN `

.**idealtwoelt2**(GEN nf, GEN x, GEN a)

Gives the valuation of the ideal x at the prime ideal *pr* in the
number field *nf*, where *pr* is in `idealprimedec`

format.
The valuation of the 0 ideal is `+oo`

.

The library syntax is `GEN `

.
Also available is
**gpidealval**(GEN nf, GEN x, GEN pr)`long `

, which returns
**idealval**(GEN nf, GEN x, GEN pr)`LONG_MAX`

if x = 0 and the valuation as a `long`

integer.

This function is deprecated, use `apply`

.

*nf* being a number field in `nfinit`

format, and x a
(row or column) vector or matrix, apply `nfalgtobasis`

to each entry
of x.

The library syntax is `GEN `

.**matalgtobasis**(GEN nf, GEN x)

This function is deprecated, use `apply`

.

*nf* being a number field in `nfinit`

format, and x a
(row or column) vector or matrix, apply `nfbasistoalg`

to each entry
of x.

The library syntax is `GEN `

.**matbasistoalg**(GEN nf, GEN x)

Let z = `Mod(A, T)`

be a polmod, and Q be its minimal
polynomial, which must satisfy deg(Q) = deg(T).
Returns a "reverse polmod" `Mod(B, Q)`

, which is a root of T.

This is quite useful when one changes the generating element in algebraic extensions:

? u = Mod(x, x^3 - x -1); v = u^5; ? w = modreverse(v) %2 = Mod(x^2 - 4*x + 1, x^3 - 5*x^2 + 4*x - 1)

which means that x^3 - 5x^2 + 4x -1 is another defining polynomial for the cubic field ℚ(u) = ℚ[x]/(x^3 - x - 1) = ℚ[x]/(x^3 - 5x^2 + 4x - 1) = ℚ(v), and that u → v^2 - 4v + 1 gives an explicit isomorphism. From this, it is easy to convert elements between the A(u) ∈ ℚ(u) and B(v) ∈ ℚ(v) representations:

? A = u^2 + 2*u + 3; subst(lift(A), 'x, w) %3 = Mod(x^2 - 3*x + 3, x^3 - 5*x^2 + 4*x - 1) ? B = v^2 + v + 1; subst(lift(B), 'x, v) %4 = Mod(26*x^2 + 31*x + 26, x^3 - x - 1)

If the minimal polynomial of z has lower degree than expected, the routine fails

? u = Mod(-x^3 + 9*x, x^4 - 10*x^2 + 1) ? modreverse(u) *** modreverse: domain error in modreverse: deg(minpoly(z)) < 4 *** Break loop: type 'break' to go back to GP prompt break> Vec( dbg_err() ) \\ ask for more info ["e_DOMAIN", "modreverse", "deg(minpoly(z))", "<", 4, Mod(-x^3 + 9*x, x^4 - 10*x^2 + 1)] break> minpoly(u) x^2 - 8

The library syntax is `GEN `

.**modreverse**(GEN z)

Gives the vector of the slopes of the Newton
polygon of the polynomial x with respect to the prime number p. The n
components of the vector are in decreasing order, where n is equal to the
degree of x. Vertical slopes occur iff the constant coefficient of x is
zero and are denoted by `+oo`

.

The library syntax is `GEN `

.**newtonpoly**(GEN x, GEN p)

Given an algebraic number x in the number field *nf*,
transforms it to a column vector on the integral basis

.*nf*.zk

? nf = nfinit(y^2 + 4); ? nf.zk %2 = [1, 1/2*y] ? nfalgtobasis(nf, [1,1]~) %3 = [1, 1]~ ? nfalgtobasis(nf, y) %4 = [0, 2]~ ? nfalgtobasis(nf, Mod(y, y^2+4)) %5 = [0, 2]~

This is the inverse function of `nfbasistoalg`

.

The library syntax is `GEN `

.**algtobasis**(GEN nf, GEN x)

Let T(X) be an irreducible polynomial with integral coefficients. This
function returns an integral basis of the number field defined by T,
that is a ℤ-basis of its maximal order. If present, `dK`

is set
to the discriminant of the returned order. The basis elements are given as
elements in K = ℚ[X]/(T), in Hermite normal form with respect to the
ℚ-basis (1,X,...,X^{deg T-1}) of K, lifted to ℚ[X].
In particular its first element is always 1 and its i-th element is a
polynomial of degree i-1 whose leading coefficient is the inverse of an
integer: the product of those integers is the index of ℤ[X]/(T) in the
maximal order ℤ_{K}:

? nfbasis(x^2 + 4) \\ Z[X]/(T) has index 2 in Z_{K}%1 = [1, x/2] ? nfbasis(x^2 + 4, &D) %2 = [1, x/2] ? D %3 = -4

This function uses a modified version of the round 4 algorithm, due to David Ford, Sebastian Pauli and Xavier Roblot.

**Local basis, orders maximal at certain primes.**

Obtaining the maximal order is hard: it requires factoring the discriminant
D of T. Obtaining an order which is maximal at a finite explicit set of
primes is easy, but it may then be a strict suborder of the maximal order. To
specify that we are interested in a given set of places only, we can replace
the argument T by an argument [T,*listP*], where *listP* encodes
the primes we are interested in: it must be a factorization matrix, a vector
of integers or a single integer.

***** Vector: we assume that it contains distinct *prime* numbers.

***** Matrix: we assume that it is a two-column matrix of a
(partial) factorization of D; namely the first column contains
distinct *primes* and the second one the valuation of D at each of
these primes.

***** Integer B: this is replaced by the vector of primes up to B. Note
that the function will use at least O(B) time: a small value, about
10^5, should be enough for most applications. Values larger than 2^{32}
are not supported.

In all these cases, the primes may or may not divide the discriminant D
of T. The function then returns a ℤ-basis of an order whose index is
not divisible by any of these prime numbers. The result may actually be
a global integral basis, in particular if all the prime divisors of the
*field* discriminant are included, but this is not guaranteed!
Note that `nfinit`

has built-in support for such a check:

? K = nfinit([T, listP]); ? nfcertify(K) \\ we computed an actual maximal order %2 = [];

The first line initializes a number field structure
incorporating `nfbasis([T, listP]`

in place of a proven integral basis.
The second line certifies that the resulting structure is correct. This
allows to create an `nf`

structure attached to the number field K =
ℚ[X]/(T), when the discriminant of T cannot be factored completely,
whereas the prime divisors of disc K are known. If present, the argument
`dK`

is set to the discriminant of the returned order, and is
equal to the field discriminant if and only if the order is maximal.

Of course, if *listP* contains a single prime number p,
the function returns a local integral basis for ℤ_{p}[X]/(T):

? nfbasis(x^2+x-1001) %1 = [1, 1/3*x - 1/3] ? nfbasis( [x^2+x-1001, [2]] ) %2 = [1, x]

The following function computes the index i_{T} of ℤ[X]/(T)
in the order generated by the ℤ-basis B:

nfbasisindex(T, B) = vecprod([denominator(pollead(Q)) | Q <- B]);

In particular, B is a basis of the maximal order
if and only if `poldisc`

(T) / i_{T}^2 is equal to the field
discriminant. More generally, this formula gives the square of index of the
order given by B in ℤ_{K}. For instance, assume that P is a vector
of prime numbers containing (at least) all prime divisors of the field
discriminant, then the following construct allows to provably compute the
field discriminant and to check whether the returned basis is actually
a basis of the maximal order

? B = nfbasis([T, P], &D); ? dK = sign(D) * vecprod([p^valuation(D,p) | p<-P]); ? dK * nfbasisindex(T, B)^2 == poldisc(T)

The variable `dK`

contains the field discriminant and
the last command returns 1 if and only if B is a ℤ-basis of the
maximal order. Of course, the `nfinit`

/ `nfcertify`

approach is
simpler, but it is also more costly.

**The Buchmann-Lenstra algorithm.**

We now complicate the picture: it is in fact allowed to include
*composite* numbers instead of primes
in `listP`

(Vector or Matrix case), provided they are pairwise coprime.
The result may still be a correct integral basis if
the field discriminant factors completely over the actual primes in the
list; again, this is not guaranteed. Adding a composite C such that C^2
*divides* D may help because when we consider C as a prime and run
the algorithm, two good things can happen: either we succeed in proving that
no prime dividing C can divide the index (without actually needing to find
those primes), or the computation exhibits a nontrivial zero divisor,
thereby factoring C and we go on with the refined factorization. (Note that
including a C such that C^2 does not divide D is useless.) If neither
happen, then the computed basis need not generate the maximal order. Here is
an example:

? B = 10^5; ? listP = factor(poldisc(T), B); \\ primes <= B dividing D + cofactor ? basis = nfbasis([T, listP], &D)

If the computed discriminant D factors completely
over the primes less than B (together with the primes contained in the
`addprimes`

table), then everything is certified: D is the field
discriminant and `basis`

generates the maximal order.
This can be tested as follows:

F = factor(D, B); P = F[,1]; E = F[,2]; for (i = 1, #P, if (P[i] > B && !isprime(P[i]), warning("nf may be incorrect")));

This is a sufficient but not a necessary condition, hence the warning, instead of an error.

The function `nfcertify`

speeds up and automates the above process:

? B = 10^5; ? nf = nfinit([T, B]); ? nfcertify(nf) %3 = [] \\ nf is unconditionally correct ? [basis, disc] = [nf.zk, nf.disc];

The library syntax is `GEN `

.**nfbasis**(GEN T, GEN *dK = NULL)

Given an algebraic number x in the number field *nf*, transforms it
into `t_POLMOD`

form.

? nf = nfinit(y^2 + 4); ? nf.zk %2 = [1, 1/2*y] ? nfbasistoalg(nf, [1,1]~) %3 = Mod(1/2*y + 1, y^2 + 4) ? nfbasistoalg(nf, y) %4 = Mod(y, y^2 + 4) ? nfbasistoalg(nf, Mod(y, y^2+4)) %5 = Mod(y, y^2 + 4)

This is the inverse function of `nfalgtobasis`

.

The library syntax is `GEN `

.**basistoalg**(GEN nf, GEN x)

*nf* being as output by
`nfinit`

, checks whether the integer basis is known unconditionally.
This is in particular useful when the argument to `nfinit`

was of the
form [T, `listP`

], specifying a finite list of primes when
p-maximality had to be proven, or a list of coprime integers to which
Buchmann-Lenstra algorithm was to be applied.

The function returns a vector of coprime composite integers. If this vector
is empty, then `nf.zk`

and `nf.disc`

are correct. Otherwise, the
result is dubious. In order to obtain a certified result, one must completely
factor each of the given integers, then `addprime`

each of their prime
factors, then check whether `nfdisc(nf.pol)`

is equal to `nf.disc`

.

The library syntax is `GEN `

.**nfcertify**(GEN nf)

Let *nf* be a number field structure attached to the field K
and let P and Q
be squarefree polynomials in K[X] in the same variable. Outputs
the simple factors of the étale K-algebra A = K[X, Y] / (P(X), Q(Y)).
The factors are given by a list of polynomials R in K[X], attached to
the number field K[X]/ (R), and sorted by increasing degree (with respect
to lexicographic ordering for factors of equal degrees). Returns an error if
one of the polynomials is not squarefree.

Note that it is more efficient to reduce to the case where P and Q are irreducible first. The routine will not perform this for you, since it may be expensive, and the inputs are irreducible in most applications anyway. In this case, there will be a single factor R if and only if the number fields defined by P and Q are linearly disjoint (their intersection is K).

The binary digits of *flag* mean

1: outputs a vector of 4-component vectors [R,a,b,k], where R ranges through the list of all possible compositums as above, and a (resp. b) expresses the root of P (resp. Q) as an element of K[X]/(R). Finally, k is a small integer such that b + ka = X modulo R.

2: assume that P and Q define number fields that are linearly disjoint: both polynomials are irreducible and the corresponding number fields have no common subfield besides K. This allows to save a costly factorization over K. In this case return the single simple factor instead of a vector with one element.

A compositum is often defined by a complicated polynomial, which it is
advisable to reduce before further work. Here is an example involving
the field K(ζ_{5}, 5^{1/10}), K = ℚ(sqrt{5}):

```
? K = nfinit(y^2-5);
? L = nfcompositum(K, x^5 - y, polcyclo(5), 1); \\ list of [R,a,b,k]
? [R, a] = L[1]; \\ pick the single factor, extract R,a (ignore b,k)
? lift(R) \\ defines the compositum
%4 = x^10 + (-5/2*y + 5/2)*x^9 + (-5*y + 20)*x^8 + (-20*y + 30)*x^7 + \
(-45/2*y + 145/2)*x^6 + (-71/2*y + 121/2)*x^5 + (-20*y + 60)*x^4 + \
(-25*y + 5)*x^3 + 45*x^2 + (-5*y + 15)*x + (-2*y + 6)
? a^5 - y \\ a fifth root of y
%5 = 0
? [T, X] = rnfpolredbest(K, R, 1);
? lift(T) \\ simpler defining polynomial for K[x]/(R)
%7 = x^10 + (-11/2*y + 25/2)
? liftall(X) \\ root of R in K[x]/(T(x))
%8 = (3/4*y + 7/4)*x^7 + (-1/2*y - 1)*x^5 + 1/2*x^2 + (1/4*y - 1/4)
? a = subst(a.pol, 'x, X); \\
````a`

in the new coordinates
? liftall(a)
%10 = (-3/4*y - 7/4)*x^7 - 1/2*x^2
? a^5 - y
%11 = 0

The main variables of P and Q must be the same and have higher priority
than that of *nf* (see `varhigher`

and `varlower`

).

The library syntax is `GEN `

.**nfcompositum**(GEN nf, GEN P, GEN Q, long flag)

Given a pseudo-matrix x, computes a
nonzero ideal contained in (i.e. multiple of) the determinant of x. This
is particularly useful in conjunction with `nfhnfmod`

.

The library syntax is `GEN `

.**nfdetint**(GEN nf, GEN x)

field discriminant of the number field defined by the integral, preferably monic, irreducible polynomial T(X). Returns the discriminant of the number field ℚ[X]/(T), using the Round 4 algorithm.

**Local discriminants, valuations at certain primes.**

As in `nfbasis`

, the argument T can be replaced by [T,*listP*],
where `listP`

is as in `nfbasis`

: a vector of pairwise coprime
integers (usually distinct primes), a factorization matrix, or a single
integer. In that case, the function returns the discriminant of an order
whose basis is given by `nfbasis(T,listP)`

, which need not be the maximal
order, and whose valuation at a prime entry in `listP`

is the same as the
valuation of the field discriminant.

In particular, if `listP`

is [p] for a prime p, we can
return the p-adic discriminant of the maximal order of ℤ_{p}[X]/(T),
as a power of p, as follows:

? padicdisc(T,p) = p^valuation(nfdisc([T,[p]]), p); ? nfdisc(x^2 + 6) %2 = -24 ? padicdisc(x^2 + 6, 2) %3 = 8 ? padicdisc(x^2 + 6, 3) %4 = 3

The following function computes the discriminant of the maximal order under the assumption that P is a vector of prime numbers containing (at least) all prime divisors of the field discriminant:

globaldisc(T, P) = { my (D = nfdisc([T, P])); sign(D) * vecprod([p^valuation(D,p) | p <-P]); } ? globaldisc(x^2 + 6, [2, 3, 5]) %1 = -24

The library syntax is

. Also available is **nfdisc**(GEN T)`GEN `

,
which returns the order basis, and where **nfbasis**(GEN T, GEN *d)`*d`

receives the order
discriminant.

Given a polynomial T with integer coefficients, return
[D, *faD*] where D is `nfdisc`

(T) and
*faD* is the factorization of |D|. All the variants `[T,listP]`

are allowed (see `??nfdisc`

), in which case *faD* is the
factorization of the discriminant underlying order (which need not be maximal
at the primes not specified by `listP`

) and the factorization may
contain large composites.

? T = x^3 - 6021021*x^2 + 12072210077769*x - 8092423140177664432; ? [D,faD] = nfdiscfactors(T); print(faD); D [3, 3; 500009, 2] %2 = -6750243002187] ? T = x^3 + 9*x^2 + 27*x - 125014250689643346789780229390526092263790263725; ? [D,faD] = nfdiscfactors(T); print(faD); D [3, 3; 1000003, 2] %4 = -27000162000243 ? [D,faD] = nfdiscfactors([T, 10^3]); print(faD) [3, 3; 125007125141751093502187, 2]

In the final example, we only get a partial factorization, which is only guaranteed correct at primes ≤ 10^3.

The function also accept number field structures, for instance as output by
`nfinit`

, and returns the field discriminant and its factorization:

? T = x^3 + 9*x^2 + 27*x - 125014250689643346789780229390526092263790263725; ? nf = nfinit(T); [D,faD] = nfdiscfactors(T); print(faD); D %2 = -27000162000243 ? nf.disc %3 = -27000162000243

The library syntax is `GEN `

.**nfdiscfactors**(GEN T)

Given two elements x and y in
*nf*, computes their sum x+y in the number field *nf*.

? nf = nfinit(1+x^2); ? nfeltadd(nf, 1, x) \\ 1 + I %2 = [1, 1]~

The library syntax is `GEN `

.**nfadd**(GEN nf, GEN x, GEN y)

Given two elements x and y in
*nf*, computes their quotient x/y in the number field *nf*.

The library syntax is `GEN `

.**nfdiv**(GEN nf, GEN x, GEN y)

Given two elements x and y in
*nf*, computes an algebraic integer q in the number field *nf*
such that the components of x-qy are reasonably small. In fact, this is
functionally identical to `round(nfdiv(`

.*nf*,x,y))

The library syntax is `GEN `

.**nfdiveuc**(GEN nf, GEN x, GEN y)

This function is obsolete, use `nfmodpr`

.

Given two elements x
and y in *nf* and *pr* a prime ideal in `modpr`

format (see
`nfmodprinit`

), computes their quotient x / y modulo the prime ideal
*pr*.

The library syntax is `GEN `

.
This function is normally useless in library mode. Project your
inputs to the residue field using **nfdivmodpr**(GEN nf, GEN x, GEN y, GEN pr)`nf_to_Fq`

, then work there.

Given two elements x and y in
*nf*, gives a two-element row vector [q,r] such that x = qy+r, q is
an algebraic integer in *nf*, and the components of r are
reasonably small.

The library syntax is `GEN `

.**nfdivrem**(GEN nf, GEN x, GEN y)

Given an element x in the number field *nf*, return
the (real or) complex embeddings of x specified by optional argument
*pl*, at the current `realprecision`

:

***** *pl* omitted: return the vector of embeddings at all r_{1}+r_{2}
places;

***** *pl* an integer between 1 and r_{1}+r_{2}: return the
i-th embedding of x, attached to the i-th root of `nf.pol`

,
i.e. `nf.roots[i]`

;

***** *pl* a vector or `t_VECSMALL`

: return the vector of embeddings; the i-th
entry gives the embedding at the place attached to the *pl*[i]-th real
root of `nf.pol`

.

? nf = nfinit('y^3 - 2); ? nf.sign %2 = [1, 1] ? nfeltembed(nf, 'y) %3 = [1.25992[...], -0.62996[...] + 1.09112[...]*I]] ? nfeltembed(nf, 'y, 1) %4 = 1.25992[...] ? nfeltembed(nf, 'y, 3) \\ there are only 2 arch. places *** at top-level: nfeltembed(nf,'y,3) *** ^ — — — — — -- *** nfeltembed: domain error in nfeltembed: index > 2

The library syntax is `GEN `

.**nfeltembed**(GEN nf, GEN x, GEN pl = NULL, long prec)

Return 1 if x is an n-th power (and set y to an n-th root if the argument is present), else return 0.

? nf = nfinit(1+x^2); ? nfeltispower(nf, -4, 4, &y) %2 = 1 ? y %3 = [-1, -1]~

The library syntax is `long `

.**nfispower**(GEN nf, GEN x, long n, GEN *y = NULL)

Return 1 if x is a square (and set y to a square root if the argument is present), else return 0.

? nf = nfinit(1+x^2); ? nfeltissquare(nf, -1, &y) %2 = 1 ? y %3 = [0, -1]~

The library syntax is `long `

.**nfissquare**(GEN nf, GEN x, GEN *y = NULL)

Given two elements x and y in
*nf*, computes an element r of *nf* of the form r = x-qy with
q and algebraic integer, and such that r is small. This is functionally
identical to
`x - nfmul(`

.*nf*,round(nfdiv(*nf*,x,y)),y)

The library syntax is `GEN `

.**nfmod**(GEN nf, GEN x, GEN y)

Given two elements x and y in *nf*, computes their product x*y
in the number field *nf*.

The library syntax is `GEN `

.**nfmul**(GEN nf, GEN x, GEN y)

This function is obsolete, use `nfmodpr`

.

Given two elements x and
y in *nf* and *pr* a prime ideal in `modpr`

format (see
`nfmodprinit`

), computes their product x*y modulo the prime ideal
*pr*.

The library syntax is `GEN `

.
This function is normally useless in library mode. Project your
inputs to the residue field using **nfmulmodpr**(GEN nf, GEN x, GEN y, GEN pr)`nf_to_Fq`

, then work there.

Returns the absolute norm of x.

The library syntax is `GEN `

.**nfnorm**(GEN nf, GEN x)

Given an element x in *nf*, and a positive or negative integer k,
computes x^k in the number field *nf*.

The library syntax is `GEN `

.
**nfpow**(GEN nf, GEN x, GEN k)`GEN `

correspond to k = -1, and
**nfinv**(GEN nf, GEN x)`GEN `

to k = 2.**nfsqr**(GEN nf,GEN x)

This function is obsolete, use `nfmodpr`

.

Given an element x in *nf*, an integer k and a prime ideal
*pr* in `modpr`

format
(see `nfmodprinit`

), computes x^k modulo the prime ideal *pr*.

The library syntax is `GEN `

.
This function is normally useless in library mode. Project your
inputs to the residue field using **nfpowmodpr**(GEN nf, GEN x, GEN k, GEN pr)`nf_to_Fq`

, then work there.

Given an ideal *id* in
Hermite normal form and an element a of the number field *nf*,
finds an element r in *nf* such that a-r belongs to the ideal
and r is small.

The library syntax is `GEN `

.**nfreduce**(GEN nf, GEN a, GEN id)

This function is obsolete, use `nfmodpr`

.

Given an element x of the number field *nf* and a prime ideal
*pr* in `modpr`

format compute a canonical representative for the
class of x modulo *pr*.

The library syntax is `GEN `

.
This function is normally useless in library mode. Project your
inputs to the residue field using **nfreducemodpr**(GEN nf, GEN x, GEN pr)`nf_to_Fq`

, then work there.

Given an element x in the number field *nf*, returns the signs of
the real embeddings of x specified by optional argument *pl*:

***** *pl* omitted: return the vector of signs at all r_{1} real places;

***** *pl* an integer between 1 and r_{1}: return the sign of the
i-th embedding of x, attached to the i-th real root of `nf.pol`

,
i.e. `nf.roots[i]`

;

***** *pl* a vector or `t_VECSMALL`

: return the vector of signs; the i-th
entry gives the sign at the real place attached to the *pl*[i]-th real
root of `nf.pol`

.

? nf = nfinit(polsubcyclo(11,5,'y)); \\ Q(cos(2 pi/11)) ? nf.sign %2 = [5, 0] ? x = Mod('y, nf.pol); ? nfeltsign(nf, x) %4 = [-1, -1, -1, 1, 1] ? nfeltsign(nf, x, 1) %5 = -1 ? nfeltsign(nf, x, [1..4]) %6 = [-1, -1, -1, 1] ? nfeltsign(nf, x, 6) \\ there are only 5 real embeddings *** at top-level: nfeltsign(nf,x,6) *** ^ — — — — — -- *** nfeltsign: domain error in nfeltsign: index > 5

The library syntax is `GEN `

.**nfeltsign**(GEN nf, GEN x, GEN pl = NULL)

Returns the absolute trace of x.

The library syntax is `GEN `

.**nftrace**(GEN nf, GEN x)

Given an element x in
*nf* and a prime ideal *pr* in the format output by
`idealprimedec`

, computes the valuation v at *pr* of the
element x. The valuation of 0 is `+oo`

.

? nf = nfinit(x^2 + 1); ? P = idealprimedec(nf, 2)[1]; ? nfeltval(nf, x+1, P) %3 = 1

This particular valuation can also be obtained using
`idealval(`

, since x is then converted to a
principal ideal.*nf*,x,*pr*)

If the y argument is present, sets y = x τ^v, where τ is a
fixed "anti-uniformizer" for *pr*: its valuation at *pr* is -1;
its valuation is 0 at other prime ideals dividing

and
nonnegative at all other primes. In other words y is the part of x
coprime to *pr*.p*pr*. If x is an algebraic integer, so is y.

? nfeltval(nf, x+1, P, &y); y %4 = [0, 1]~

For instance if x = ∏_{i} x_{i}^{ei} is known to be coprime to *pr*,
where the x_{i} are algebraic integers and e_{i} ∈ ℤ then,
if v_{i} = `nfeltval`

(*nf*, x_{i}, *pr*, &y_{i}), we still
have x = ∏_{i} y_{i}^{ei}, where the y_{i} are still algebraic integers
but now all of them are coprime to *pr*. They can then be mapped to
the residue field of *pr* more efficiently than if the product had
been expanded beforehand: we can reduce mod *pr* after each ring
operation.

The library syntax is `GEN `

.
Also available are
**gpnfvalrem**(GEN nf, GEN x, GEN pr, GEN *y = NULL)`long `

, which returns
**nfvalrem**(GEN nf, GEN x, GEN pr, GEN *y = NULL)`LONG_MAX`

if x = 0 and the valuation as a `long`

integer,
and `long `

, which only returns the
valuation (y = **nfval**(GEN nf, GEN x, GEN pr)`NULL`

).

Factorization of the univariate
polynomial (or rational function) T over the number field *nf* given
by `nfinit`

; T has coefficients in *nf* (i.e. either scalar,
polmod, polynomial or column vector). The factors are sorted by increasing
degree.

The main variable of *nf* must be of *lower*
priority than that of T, see Section se:priority. However if
the polynomial defining the number field occurs explicitly in the
coefficients of T as modulus of a `t_POLMOD`

or as a `t_POL`

coefficient, its main variable must be *the same* as the main variable
of T. For example,

? nf = nfinit(y^2 + 1); ? nffactor(nf, x^2 + y); \\ OK ? nffactor(nf, x^2 + Mod(y, y^2+1)); \\ OK ? nffactor(nf, x^2 + Mod(z, z^2+1)); \\ WRONG

It is possible to input a defining polynomial for *nf*
instead, but this is in general less efficient since parts of an `nf`

structure will then be computed internally. This is useful in two
situations: when you do not need the `nf`

elsewhere, or when you cannot
initialize an `nf`

due to integer factorization difficulties when
attempting to compute the field discriminant and maximal order. In all
cases, the function runs in polynomial time using Belabas's variant
of van Hoeij's algorithm, which copes with hundreds of modular factors.

**Caveat.** `nfinit([T, listP])`

allows to compute in polynomial
time a conditional *nf* structure, which sets `nf.zk`

to an order
which is not guaranteed to be maximal at all primes. Always either use
`nfcertify`

first (which may not run in polynomial time) or make sure
to input `nf.pol`

instead of the conditional *nf*: `nffactor`

is
able to recover in polynomial time in this case, instead of potentially
missing a factor.

The library syntax is `GEN `

.**nffactor**(GEN nf, GEN T)

Gives back the *nf* element corresponding to a factorization.
The integer 1 corresponds to the empty factorization.

If e is present, e and f must be vectors of the same length (e being
integral), and the corresponding factorization is the product of the
f[i]^{e[i]}.

If not, and f is vector, it is understood as in the preceding case with e a vector of 1s: we return the product of the f[i]. Finally, f can be a regular factorization matrix.

? nf = nfinit(y^2+1); ? nffactorback(nf, [3, y+1, [1,2]~], [1, 2, 3]) %2 = [12, -66]~ ? 3 * (I+1)^2 * (1+2*I)^3 %3 = 12 - 66*I

The library syntax is `GEN `

.**nffactorback**(GEN nf, GEN f, GEN e = NULL)

This routine is obsolete, use `nfmodpr`

and `factormod`

.

Factors the univariate polynomial Q modulo the prime ideal *pr* in
the number field *nf*. The coefficients of Q belong to the number
field (scalar, polmod, polynomial, even column vector) and the main variable
of *nf* must be of lower priority than that of Q (see
Section se:priority). The prime ideal *pr* is either in
`idealprimedec`

or (preferred) `modprinit`

format. The coefficients
of the polynomial factors are lifted to elements of *nf*:

? K = nfinit(y^2+1); ? P = idealprimedec(K, 3)[1]; ? nffactormod(K, x^2 + y*x + 18*y+1, P) %3 = [x + (2*y + 1) 1] [x + (2*y + 2) 1] ? P = nfmodprinit(K, P); \\ convert to nfmodprinit format ? nffactormod(K, x^2 + y*x + 18*y+1) %5 = [x + (2*y + 1) 1] [x + (2*y + 2) 1]

Same result, of course, here about 10% faster due to the precomputation.

The library syntax is `GEN `

.**nffactormod**(GEN nf, GEN Q, GEN pr)

Let *nf* be a
number field as output by `nfinit`

, and let *aut* be a Galois
automorphism of *nf* expressed by its image on the field generator
(such automorphisms can be found using `nfgaloisconj`

). The function
computes the action of the automorphism *aut* on the object x in the
number field; x can be a number field element, or an ideal (possibly
extended). Because of possible confusion with elements and ideals, other
vector or matrix arguments are forbidden.

? nf = nfinit(x^2+1); ? L = nfgaloisconj(nf) %2 = [-x, x]~ ? aut = L[1]; /* the nontrivial automorphism */ ? nfgaloisapply(nf, aut, x) %4 = Mod(-x, x^2 + 1) ? P = idealprimedec(nf,5); /* prime ideals above 5 */ ? nfgaloisapply(nf, aut, P[2]) == P[1] %6 = 0 \\ !!!! ? idealval(nf, nfgaloisapply(nf, aut, P[2]), P[1]) %7 = 1

The surprising failure of the equality test (`%7`

) is
due to the fact that although the corresponding prime ideals are equal, their
representations are not. (A prime ideal is specified by a uniformizer, and
there is no guarantee that applying automorphisms yields the same elements
as a direct `idealprimedec`

call.)

The automorphism can also be given as a column vector, representing the
image of `Mod(x, nf.pol)`

as an algebraic number. This last
representation is more efficient and should be preferred if a given
automorphism must be used in many such calls.

? nf = nfinit(x^3 - 37*x^2 + 74*x - 37); ? aut = nfgaloisconj(nf)[2]; \\ an automorphism in basistoalg form %2 = -31/11*x^2 + 1109/11*x - 925/11 ? AUT = nfalgtobasis(nf, aut); \\ same in algtobasis form %3 = [16, -6, 5]~ ? v = [1, 2, 3]~; nfgaloisapply(nf, aut, v) == nfgaloisapply(nf, AUT, v) %4 = 1 \\ same result... ? for (i=1,10^5, nfgaloisapply(nf, aut, v)) time = 463 ms. ? for (i=1,10^5, nfgaloisapply(nf, AUT, v)) time = 343 ms. \\ but the latter is faster

The library syntax is `GEN `

.**galoisapply**(GEN nf, GEN aut, GEN x)

*nf* being a number field as output by `nfinit`

, computes the
conjugates of a root r of the nonconstant polynomial x = *nf*[1]
expressed as polynomials in r. This also makes sense when the number field
is not Galois since some conjugates may lie in the field.
*nf* can simply be a polynomial.

If no flags or *flag* = 0, use a combination of flag 4 and 1 and the result
is always complete. There is no point whatsoever in using the other flags.

If *flag* = 1, use `nfroots`

: a little slow, but guaranteed to work in
polynomial time.

If *flag* = 4, use `galoisinit`

: very fast, but only applies to (most)
Galois fields. If the field is Galois with weakly super-solvable Galois
group (see `galoisinit`

), return the complete list of automorphisms, else
only the identity element. If present, d is assumed to be a multiple of the
least common denominator of the conjugates expressed as polynomial in a root
of *pol*.

This routine can only compute ℚ-automorphisms, but it may be used to get K-automorphism for any base field K as follows:

rnfgaloisconj(nfK, R) = \\ K-automorphisms of L = K[X] / (R) { my(polabs, N,al,S, ala,k, vR); R *= Mod(1, nfK.pol); \\ convert coeffs to polmod elts of K vR = variable(R); al = Mod(variable(nfK.pol),nfK.pol); [polabs,ala,k] = rnfequation(nfK, R, 1); Rt = if(k==0,R,subst(R,vR,vR-al*k)); N = nfgaloisconj(polabs) % Rt; \\ Q-automorphisms of L S = select(s->subst(Rt, vR, Mod(s,Rt)) == 0, N); if (k==0, S, apply(s->subst(s,vR,vR+k*al)-k*al,S)); } K = nfinit(y^2 + 7); rnfgaloisconj(K, x^4 - y*x^3 - 3*x^2 + y*x + 1) \\ K-automorphisms of L

The library syntax is `GEN `

.
Use directly
**galoisconj0**(GEN nf, long flag, GEN d = NULL, long prec)`GEN `

, corresponding to **galoisconj**(GEN nf, GEN d)*flag* = 0, the others
only have historical interest.

Given *nf* a number field in *nf* or *bnf* format,
a `t_VEC`

*Lpr* of primes of *nf* and a `t_VEC`

*Ld* of
positive integers of the same length, a `t_VECSMALL`

*pl* of length
r_{1} the number of real places of *nf*, computes a polynomial with
coefficients in *nf* defining a cyclic extension of *nf* of
minimal degree satisfying certain local conditions:

***** at the prime Lpr[i], the extension has local degree a multiple
of Ld[i];

***** at the i-th real place of *nf*, it is complex if pl[i] = -1
(no condition if pl[i] = 0).

The extension has degree the LCM of the local degrees. Currently, the degree
is restricted to be a prime power for the search, and to be prime for the
construction because of the `rnfkummer`

restrictions.

When *nf* is ℚ, prime integers are accepted instead of `prid`

structures. However, their primality is not checked and the behavior is
undefined if you provide a composite number.

**Warning.** If the number field *nf* does not contain the n-th
roots of unity where n is the degree of the extension to be computed,
the function triggers the computation of the *bnf* of nf(ζ_{n}),
which may be costly.

? nf = nfinit(y^2-5); ? pr = idealprimedec(nf,13)[1]; ? pol = nfgrunwaldwang(nf, [pr], [2], [0,-1], 'x) %3 = x^2 + Mod(3/2*y + 13/2, y^2 - 5)

The library syntax is `GEN `

where **nfgrunwaldwang**(GEN nf, GEN Lpr, GEN Ld, GEN pl, long v = -1)`v`

is a variable number.

If *pr* is omitted,
compute the global quadratic Hilbert symbol (a,b) in *nf*, that
is 1 if x^2 - a y^2 - b z^2 has a non trivial solution (x,y,z) in
*nf*, and -1 otherwise. Otherwise compute the local symbol modulo
the prime ideal *pr*, as output by `idealprimedec`

.

The library syntax is `long `

.**nfhilbert0**(GEN nf, GEN a, GEN b, GEN pr = NULL)

Also available is `long `

(global
quadratic Hilbert symbol), where **nfhilbert**(GEN nf,GEN a,GEN b)`nf`

is a true *nf* structure.

Given a pseudo-matrix (A,I), finds a
pseudo-basis (B,J) in Hermite normal form of the module it generates.
If *flag* is nonzero, also return the transformation matrix U such that
AU = [0|B].

The library syntax is `GEN `

.
Also available:**nfhnf0**(GEN nf, GEN x, long flag)

`GEN `

(**nfhnf**(GEN nf, GEN x)*flag* = 0).

`GEN `

simplifies the pseudo-basis
x = (A,I), returning a pseudo-basis (B,J). The ideals in the list J
are integral, primitive and either trivial (equal to the full ring of
integer) or nonprincipal.**rnfsimplifybasis**(GEN bnf, GEN x)

Given a pseudo-matrix (A,I)
and an ideal *detx* which is contained in (read integral multiple of) the
determinant of (A,I), finds a pseudo-basis in Hermite normal form
of the module generated by (A,I). This avoids coefficient explosion.
*detx* can be computed using the function `nfdetint`

.

The library syntax is `GEN `

.**nfhnfmod**(GEN nf, GEN x, GEN detx)

*pol* being a nonconstant irreducible polynomial in ℚ[X],
preferably monic and integral, initializes a
*number field* (or *nf*) structure attached to the field K defined
by *pol*. As such, it's a technical object passed as the first argument
to most `nf`

*xxx* functions, but it contains some information which
may be directly useful. Access to this information via *member
functions* is preferred since the specific data organization given below
may change in the future. Currently, `nf`

is a row vector with 9
components:

*nf*[1] contains the polynomial *pol* (

).*nf*.pol

*nf*[2] contains [r1,r2] (

, *nf*.sign

,
*nf*.r1

), the number of real and complex places of K.*nf*.r2

*nf*[3] contains the discriminant d(K) (

) of K.*nf*.disc

*nf*[4] contains the index of *nf*[1] (

),
i.e. [ℤ*nf*.index_{K} : ℤ[θ]], where θ is any root of *nf*[1].

*nf*[5] is a vector containing 7 matrices M, G, *roundG*, T,
*MD*, *TI*, *MDI* and a vector *vP* defined as follows:

***** M is the (r1+r2) x n matrix whose columns represent
the numerical values of the conjugates of the elements of the integral
basis.

***** G is an n x n matrix such that T2 = ^t G G,
where T2 is the quadratic form T_{2}(x) = ∑ |σ(x)|^2, σ
running over the embeddings of K into ℂ.

***** *roundG* is a rescaled copy of G, rounded to nearest
integers.

***** T is the n x n matrix whose coefficients are
Tr(ω_{i}ω_{j}) where the ω_{i} are the elements of the
integral basis. Note also that det(T) is equal to the discriminant of the
field K. Also, when understood as an ideal, the matrix T^{-1}
generates the codifferent ideal.

***** The columns of MD (

) express a ℤ-basis
of the different of K on the integral basis.*nf*.diff

***** *TI* is equal to the primitive part of T^{-1}, which has
integral coefficients.

***** *MDI* is a two-element representation (for faster
ideal product) of d(K) times the codifferent ideal
(

, which is an integral ideal). This is
used in *nf*.disc**nf*.codiff`idealinv`

.

***** *vP* is the list of prime divisors of the field discriminant,
i.e, the ramified primes (

); *nf*.p`nfdiscfactors(nf)`

is the
preferred way to access that information.

*nf*[6] is the vector containing the r1+r2 roots
(

) of *nf*.roots*nf*[1] corresponding to the r1+r2
embeddings of the number field into ℂ (the first r1 components are real,
the next r2 have positive imaginary part).

*nf*[7] is a ℤ-basis for dℤ_{K}, where d = [ℤ_{K}:ℤ(θ)],
expressed on the powers of θ. The multiplication by
d ensures that all polynomials have integral coefficients
and *nf*[7] / d (

) is an integral basis for ℤ*nf*.zk_{K}.
Its first element is guaranteed to be 1. This basis is LLL-reduced with
respect to T_{2} (strictly speaking, it is a permutation of such a basis, due
to the condition that the first element be 1).

*nf*[8] is the n x n integral matrix expressing the power
basis in terms of the integral basis, and finally

*nf*[9] is the n x n^2 matrix giving the multiplication table
of the integral basis.

If a non monic or non integral polynomial is input, `nfinit`

will
transform it, and return a structure attached to the new (monic integral)
polynomial together with the attached change of variables, see *flag* = 3.
It is allowed, though not very useful given the existence of
`nfnewprec`

, to input a *nf* or a *bnf* instead of a polynomial.
It is also allowed to input a *rnf*, in which case an `nf`

structure
attached to the absolute defining polynomial `polabs`

is returned (*flag* is
then ignored).

? nf = nfinit(x^3 - 12); \\ initialize number field Q[X] / (X^3 - 12) ? nf.pol \\ defining polynomial %2 = x^3 - 12 ? nf.disc \\ field discriminant %3 = -972 ? nf.index \\ index of power basis order in maximal order %4 = 2 ? nf.zk \\ integer basis, lifted to Q[X] %5 = [1, x, 1/2*x^2] ? nf.sign \\ signature %6 = [1, 1] ? factor(abs(nf.disc )) \\ determines ramified primes %7 = [2 2] [3 5] ? idealfactor(nf, 2) %8 = [[2, [0, 0, -1]~, 3, 1, [0, 1, 0]~] 3] \\ 𝔭_{2}^3

**Huge discriminants, helping nfdisc.**

In case *pol* has a huge discriminant which is difficult to factor,
it is hard to compute from scratch the maximal order. The following
special input formats are also accepted:

***** [*pol*, B] where *pol* is a monic integral polynomial and
B is the lift of an integer basis, as would be computed by `nfbasis`

:
a vector of polynomials with first element 1 (implicitly modulo *pol*).
This is useful if the maximal order is known in advance.

***** [*pol*, B, P] where *pol* and B are as above
(a monic integral polynomial and the lift of an integer basis), and P is
the list of ramified primes in the extension.

***** [*pol*, `listP`

] where *pol* is a rational polynomial and
`listP`

specifies a list of primes as in `nfbasis`

. Instead of the
maximal order, `nfinit`

then computes
an order which is maximal at these particular primes as well as the primes
contained in the private prime table, see `addprimes`

. The result has
a good chance of being correct when the discriminant `nf.disc`

factors
completely over this set of primes but this is not guaranteed. The function
`nfcertify`

automates this:

? pol = polcompositum(x^5 - 101, polcyclo(7))[1]; ? nf = nfinit( [pol, 10^3] ); ? nfcertify(nf) %3 = []

A priori, `nf.zk`

defines an order which is only known
to be maximal at all primes ≤ 10^3 (no prime ≤ 10^3 divides
`nf.index`

). The certification step proves the correctness of the
computation. Had it failed, that particular `nf`

structure could
not have been trusted and may have caused routines using it to fail randomly.
One particular function that remains trustworthy in all cases is
`idealprimedec`

when applied to a prime included in the above list
of primes or, more generally, a prime not dividing any entry in
`nfcertify`

output.

In order to explain the meaning of *flag*, let P =
`polredbest`

(*pol*), a polynomial defining the same number field
obtained using the LLL algorithm on the lattice (ℤ_{K}, T_{2}), which may be
equal to *pol* but is usually different and simpler. Binary digits of
*flag* mean:

***** 1: return [*nf*,`Mod`

(a,P)], where *nf* is
`nfinit`

(P) and `Mod`

(a,P) = `Mod`

(x,*pol*) gives the
change of variables. If only this bit is set, the behaviour is useless since
we have P = *pol*.

***** 2: return `nfinit`

(P).

Both flags are set automatically when *pol* is not monic or not
integral: first a linear change of variables is performed, to get a monic
integral polynomial, then `polredbest`

.

***** 4: do not LLL-reduce `nf.zk`

, which saves time in large degrees,
you may expect to gain a factor 2 or so in degree n ≥ 100 or more, at
the expense of *possibly* slowing down later uses of the *nf*
structure. Use this flag if you only need basic arithmetic
(the `nfelt*`

, `nfmodpr*`

and `ideal*`

functions); or if you
expect the natural basis of the maximal order to contain small elements, this
will be the case for cyclotomic fields for instance. On the other hand,
functions involving LLL reduction of rank
n lattices should be avoided since each call will be about as costly as the
initial LLL reduction that the flag prevents and may become more costly
because of this missing initial reduction. In particular it is silly to use
this flag in addition to the first two, although GP will not protest.

? T = polcyclo(307); ? K = nfinit(T); time = 19,390 ms. ? a = idealhnf(K,1-x); time = 477ms ? idealfactor(K, a) time = 294ms ? Kno = nfinit(T, 4); time = 11,256 ms. ? ano = idealhnf(Kno,1-x); \\ no slowdown, even sligthly faster time = 460ms ? idealfactor(Kno, ano) time = 264ms ? nfinit(T, 2); \\ polredbest is very slow in high degree time = 4min, 34,870 ms. ? norml2(%.pol) == norml2(T) \\ and gains nothing here %9 = 1

The library syntax is `GEN `

.
Also available are
**nfinit0**(GEN pol, long flag, long prec)`GEN `

(**nfinit**(GEN x, long prec)*flag* = 0),
`GEN `

(**nfinitred**(GEN x, long prec)*flag* = 2),
`GEN `

(**nfinitred2**(GEN x, long prec)*flag* = 3).
Instead of the above hardcoded numerical flags in `nfinit0`

, one should
rather use an or-ed combination of

***** `nf_RED`

: find a simpler defining polynomial,

***** `nf_ORIG`

: also return the change of variable,

***** `nf_NOLLL`

: do not LLL-reduce the maximal order ℤ-basis.

Returns 1 if x is an ideal in the number field *nf*, 0 otherwise.

The library syntax is `long `

.**isideal**(GEN nf, GEN x)

Let f and g define number fields, where f and g are irreducible
polynomials in ℚ[X] and *nf* structures as output by `nfinit`

.
If either f or g is not irreducible, the result is undefined.
Tests whether the number field f is conjugate to a subfield of the field
g. If not, the output is the integer 0; if it is, the output depends on
the value of *flag*:

***** *flag* = 0 (default): return a vector of polynomials [a_{1},...,a_{n}]
with rational coefficients, representing all distinct embeddings: we have
g | f o a_{i} for all i.

***** *flag* = 1: return a single polynomial a representing a single
embedding; this can be n times faster than the default when the
embeddings have huge coefficients.

***** *flag* = 2: return a vector of rational functions [r_{1},...,r_{n}]
whose denominators are coprime to g and such that r_{i} % g is the
polynomial a_{i} from *flag* = 0. This variant is always faster than *flag* = 0
but produces results which are harder to use. If the denominators are hard to
invert in ℚ[X]/(g), this may be even faster than *flag* = 1.

? T = x^6 + 3*x^4 - 6*x^3 + 3*x^2 + 18*x + 10; ? U = x^3 + 3*x^2 + 3*x - 2 ? nfisincl(U, T) %3 = [24/179*x^5-27/179*x^4+80/179*x^3-234/179*x^2+380/179*x+94/179] ? a = nfisincl(U, T, 1) %4 = 24/179*x^5-27/179*x^4+80/179*x^3-234/179*x^2+380/179*x+94/179 ? subst(U, x, Mod(a,T)) %5 = Mod(0, x^6 + 3*x^4 - 6*x^3 + 3*x^2 + 18*x + 10) ? nfisincl(U, T, 2) \\ a as a t_RFRAC %6 = [(2*x^3 - 3*x^2 + 2*x + 4)/(3*x^2 - 1)] ? (a - %[1]) % T %7 = 0 ? #nfisincl(x^2+1, T) \\ two embeddings %8 = 2 \\ same result with nf structures ? L = nfinit(T); K = nfinit(U); v = [a]; ? nfisincl(U, L) == v %10 = 1 ? nfisincl(K, T) == v %11 = 1 ? nfisincl(K, L) == v %12 = 1 \\ comparative bench: an nf is a little faster, esp. for the subfield ? B = 2000; ? for (i=1, B, nfisincl(U,T)) time = 1,364 ms. ? for (i=1, B, nfisincl(K,T)) time = 988 ms. ? for (i=1, B, nfisincl(U,L)) time = 1,341 ms. ? for (i=1, B, nfisincl(K,L)) time = 880 ms.

Using an *nf* structure for the tentative subfield is
faster if the structure is already available. On the other hand, the gain in
`nfisincl`

is usually not sufficient to make it worthwhile to initialize
only for that purpose.

? for (i=1, B, nfinit(U)) time = 590 ms.

A final more complicated example

? f = x^8 - 72*x^6 + 1944*x^4 - 30228*x^2 - 62100*x - 34749; ? g = nfsplitting(f); poldegree(g) %2 = 96 ? #nfisincl(f, g) time = 559 ms. %3 = 8 ? nfisincl(f,g,1); time = 172 ms. ? v = nfisincl(f,g,2); time = 199 ms. ? apply(x->poldegree(denominator(x)), v) %6 = [81, 81, 81, 81, 81, 81, 80, 81] ? v % g; time = 407 ms.

This final example shows that mapping rational functions to
ℚ[X]/(g) can be more costly than that the rest of the algorithm. Note that
`nfsplitting`

also admits a *flag* yielding an embedding.

The library syntax is `GEN `

.
Also available is
**nfisincl0**(GEN f, GEN g, long flag)`GEN `

(**nfisisom**(GEN a, GEN b)*flag* = 0).

As `nfisincl`

, but tests for isomorphism. More efficient if
f or g is a number field structure.

? f = x^6 + 30*x^5 + 495*x^4 + 1870*x^3 + 16317*x^2 - 22560*x + 59648; ? g = x^6 + 42*x^5 + 999*x^4 + 8966*x^3 + 36117*x^2 + 21768*x + 159332; ? h = x^6 + 30*x^5 + 351*x^4 + 2240*x^3 + 10311*x^2 + 35466*x + 58321; ? #nfisisom(f,g) \\ two isomorphisms %3 = 2 ? nfisisom(f,h) \\ not isomorphic %4 = 0 \\ comparative bench ? K = nfinit(f); L = nfinit(g); B = 10^3; ? for (i=1, B, nfisisom(f,g)) time = 6,124 ms. ? for (i=1, B, nfisisom(K,g)) time = 3,356 ms. ? for (i=1, B, nfisisom(f,L)) time = 3,204 ms. ? for (i=1, B, nfisisom(K,L)) time = 3,173 ms.

The function is usually very fast when the fields are nonisomorphic, whenever the fields can be distinguished via a simple invariant such as degree, signature or discriminant. It may be slower when the fields share all invariants, but still faster than computing actual isomorphisms:

\\ usually very fast when the answer is 'no': ? for (i=1, B, nfisisom(f,h)) time = 32 ms. \\ but not always ? u = x^6 + 12*x^5 + 6*x^4 - 377*x^3 - 714*x^2 + 5304*x + 15379 ? v = x^6 + 12*x^5 + 60*x^4 + 166*x^3 + 708*x^2 + 6600*x + 23353 ? nfisisom(u,v) %13 = 0 ? polsturm(u) == polsturm(v) %14 = 1 ? nfdisc(u) == nfdisc(v) %15 = 1 ? for(i=1,B, nfisisom(u,v)) time = 1,821 ms. ? K = nfinit(u); L = nfinit(v); ? for(i=1,B, nfisisom(K,v)) time = 232 ms.

The library syntax is `GEN `

.**nfisisom**(GEN f, GEN g)

Let *nf* be a *nf* structure attached to a number field K,
let a ∈ K and let *pr* be a *prid* structure attached to a
maximal ideal v. Return 1 if a is an n-th power in the completed
local field K_{v}, and 0 otherwise.

? K = nfinit(y^2+1); ? P = idealprimedec(K,2)[1]; \\ the ramified prime above 2 ? nfislocalpower(K,P,-1, 2) \\ -1 is a square %3 = 1 ? nfislocalpower(K,P,-1, 4) \\ ... but not a 4-th power %4 = 0 ? nfislocalpower(K,P,2, 2) \\ 2 is not a square %5 = 0 ? Q = idealprimedec(K,5)[1]; \\ a prime above 5 ? nfislocalpower(K,Q, [0, 32]~, 30) \\ 32*I is locally a 30-th power %7 = 1

The library syntax is `long `

.**nfislocalpower**(GEN nf, GEN pr, GEN a, GEN n)

This function is obsolete, use `nfmodpr`

.

Kernel of the matrix a in ℤ_{K}/*pr*, where *pr* is in
**modpr** format (see `nfmodprinit`

).

The library syntax is `GEN `

.
This function is normally useless in library mode. Project your
inputs to the residue field using **nfkermodpr**(GEN nf, GEN x, GEN pr)`nfM_to_FqM`

, then work there.

Find number fields (up to isomorphism) with Galois group of Galois closure isomorphic to G with s complex places. This function supports the following groups:

***** degree 2: C_{2} = 2T1;

***** degree 3: C_{3} = 3T1 and S_{3} = 3T2;

***** degree 4: C_{4} = 4T1, V_{4} = 4T2, D_{4} = 4T3, A_{4} = 4T4, and S_{4} = 4T5;

***** degree 5: C_{5} = 5T1, D_{5} = 5T2, F_{5} = M_{20} = 5T3, and A_{5} = 5T4;

***** degree 6: C_{6} = 6T1, S_{3}(6) = D_{6}(6) = 6T2, D_{6}(12) = 6T3,
A_{4}(6) = 6T4, S_{3} x C_{3} = 6T5, A_{4}(6) x C_{2} = 6T6, S_{4}(6)^ += 6T7,
S_{4}(6)^ -= 6T8, S_{3}^2 = 6T9, C_{3}^2:C_{4} = 6T10, S_{4}(6) x C_{2} = 6T11,
A_{5}(6) = PSL_{2}(5) = 6T12, and C_{3}^2:D_{4} = 6T13;

***** degree 7: C_{7} = 7T1, D_{7} = 7T2, M_{21} = 7T3, and M_{42} = 7T4;

***** degree 9: C_{9} = 9T1, C_{3} x C_{3} = 9T2, and D_{9} = 9T3;

***** degree ℓ with ℓ prime: C_ℓ = ℓ T1 and D_ℓ = ℓ T2.

The groups A_{5} and A5(6) require the optional package `nflistdata`

.

In addition, if N is a polynomial, all transitive subgroups of S_{n}
with n ≤ 15, as well as alternating groups A_{n} and the full symmetric group
S_{n} for all n (see below for details and explanations).

The groups are coded as [n,k] using the `nTk`

format where n is the
degree and k is the T-number, the index in the classification of
transitive subgroups of S_{n}.

Alternatively, the groups C_{n}, D_{n},
A_{n}, S_{n}, V_{4}, F_{5} = M_{20}, M_{21} and M_{42} can be input as
character strings exactly as written, lifting subscripts; for instance
`"S4"`

or `"M21"`

. If the group is not recognized or is
unsupported the function raises an exception.

The number fields are computed on the fly (and not from a preexisting table)
using a variety of algorithms, with the exception of A_{5} and A_{5}(6) which
are obtained by table lookup.
The algorithms are recursive and use the following ingredients: build
distinguished subfields (or resolvent fields in Galois closures) of smaller
degrees, use class field theory to build abelian extensions over a known
base, select subfields using Galois theory. Because of our use of class
field theory, and ultimately `bnfinit`

, all results depend on the GRH in
degree n > 3.

To avoid wasting time, the output polynomials defining the number fields are
usually not the simplest possible, use `polredbest`

or `polredabs`

to reduce them.

The non-negative integer s specifies the number of complex places, between 0 and n/2. Additional supported values are:

***** s = -1 (default) all signatures;

***** s = -2 all signatures, given by increasing number of complex
places; in degree n, this means a vector with 1 + floor(n/2)
components: the i-th entry corresponds to s = i - 1.

If the irreducible monic polynomial F ∈ ℤ[X] is specified, give only
number fields having ℚ[X]/(F) as a subfield, or in the case of
S_{3}, D_ℓ, A_{4}, S_{4}, F_{5}, M_{21} and M_{42}, as a resolvent
field (see also the function `nfresolvent`

for these cases).

The parameter N can be the following:

***** a positive integer: find all fields with absolute discriminant N
(recall that the discriminant over ℚ is (-1)^s N).

***** a pair of non-negative real numbers [a,b] specifying a real interval:
find all fields with absolute value of discriminant between a and b.
For most Galois groups, this is faster than iterating on individual N.

***** omitted (default): a few fields of small discriminant (not always
those with smallest absolute discriminant) are output with given G
and s; usually about 10, less if too difficult to find. The parameter
F is ignored.

***** a polynomial with main variable, say t, of priority lower than x.
The program outputs a *regular* polynomial in ℚ(t)[x] (in fact in
ℤ[x,t]) with the given Galois group. By Hilbert irreducibility, almost all
specializations of t will give suitable polynomials. The parameters s and
F are ignored. This is implemented for all transitive subgroups of
S_{n} with n ≤ 15 as well as for the alternating and symmetric groups A_{n} and
S_{n} for all n. Polynomials for A_{n} were inspired by J.-F. Mestre,
a few polynomials in degree ≤ 8 come from G. W. Smith,
"Some polynomials over ℚ(t) and their
Galois groups", *Math. Comp.*, **69** (230), 1999, pp. 775--796
most others in degree ≤ 11 were provided by J. Klüners and G. Malle
(see G. Malle and B. H. Matzat, *Inverse Galois Theory*, Springer,
1999) and T. Dokchitser completed the list up to degree 15. But for
A_{n} and S_{n}, subgroups of S_{n} for n > 7 require the optional
`nflistdata`

package.

**Complexity.** : For a positive integer N, the complexity is
subexponential in log N (and involves factoring N). For an interval
[a,b], the complexity is roughly as follows, ignoring terms which are
subexponential in log b. It is usually linear in the output size.

***** C_{n}: O(b^{1/φ(n)}) for n = 2, 4, 6, 9 or any odd prime;

***** D_{n}: O(b^{2/φ(n)}) for n = 4 or any odd prime;

***** V_{4}, A_{4}: O(b^{1/2}), S_{4}: O(b);
N.B. The subexponential terms are expensive for A_{4} and S_{4}.

***** M_{20}: O(b).

***** S_{4}(6)^-, S_{4}(6)^+ A_{4}(6) x C_{2}, S_{3} x S_{3},
S_{4}(6) x C_{2} : O(b),
D_{6}(12), A_{4}(6), S_{3}(6), S_{3} x C_{3}, C_{3}^2:C_{4}: O(b^{1/2}).

***** M_{21}, M_{42}: O(b).

***** C_{3} x C_{3}: O(b^{1/3}), D_{9}: O(b^{5/12}).

? #nflist("S3", [1, 10^5]) \\ S3 cubic fields %1 = 21794 ? #nflist("S3", [1, 10^5], 0) \\ real S3 cubic fields (0 complex place) %2 = 4753 ? #nflist("S3", [1, 10^5], 1) \\ complex cubic fields (1 complex place) %3 = 17041 ? v = nflist("S3", [1, 10^5], -2); apply(length,v) %4 = [4753, 17041] ? nflist("S4") \\ a few S4 fields %5 = [x^4 + 12*x^2 - 8*x + 16, x^4 - 2*x^2 - 8*x + 25, ...] ? nflist("S4",,0) \\ a few real S4 fields %6 = [x^4 - 52*x^2 - 56*x + 48, x^4 - 26*x^2 - 8*x + 1, ...] ? nflist("S4",,-2) \\ a few real S4 fields, by signature %7 = [[x^4 - 52*x^2 - 56*x + 48, ...], [x^4 - 8*x - 16, ... ], [x^4 + 138*x^2 - 8*x + 4541, ...]] ? nflist("S3",,,x^2+23) \\ a few cubic fields with resolvent Q(sqrt(-23)) %8 = [x^3 + x + 1, x^3 + 2*x + 1, ...] ? nflist("C3", 3969) \\ C3 fields of given discriminant %9 = [x^3 - 21*x + 28, x^3 - 21*x - 35] ? nflist([3,1], 3969) \\ C3 fields, using nTt label %10 = [x^3 - 21*x + 28, x^3 - 21*x - 35] ? P = nflist([8,12],t) \\ geometric 8T12 polynomial %11 = x^8 - 22*t*x^6 + 135*t^2*x^4 - 150*t^3*x^2 + t^4 ? polgalois(subst(P, t, 11)) %12 = [24, 1, 12, "2A_{4}(8)=[2]A(4)=SL(2,3)"] ? nflist("S11") *** at top-level: nflist("S11") *** ^ — — — — - *** nflist: unsupported group (S11). Use one of "C1"=[1,1]; "C2"=[2,1]; "C3"=[3,1], "S3"=[3,2]; "C4"=[4,1], "V4"=[4,2], "D4"=[4,3], "A4"=[4,4], "S4"=[4,5]; "C5"=[5,1], "D5"=[5,2], "F5"="M20"=[5,3], "A5"=[5,4]; "C6"=[6,1], "D6"=[6,2], [6,3], ..., [6,13]; "C7"=[7,1], "D7"=[7,2], "M21"=[7,3], "M42"=[7,4]; "C9"=[9,1], [9,2], "D9"=[9,3]." Also supported are "Cp"=[p,1] and "Dp"=[p,2] for any odd prime p. ? nflist("S25", 't) %13 = x^25 + x*t + 1

The library syntax is `GEN `

.**nflist**(GEN G, GEN N = NULL, long s, GEN F = NULL)

Map x to a `t_FFELT`

in the residue field modulo *pr*.
The argument *pr* is either a maximal ideal in `idealprimedec`

format or, preferably, a *modpr* structure from `nfmodprinit`

. The
function `nfmodprlift`

allows to lift back to ℤ_{K}.

Note that the function applies to number field elements and not to
vector / matrices / polynomials of such. Use `apply`

to convert
recursive structures.

? K = nfinit(y^3-250); ? P = idealprimedec(K, 5)[2]; ? modP = nfmodprinit(K, P, 't); ? K.zk %4 = [1, 1/5*y, 1/25*y^2] ? apply(t->nfmodpr(K,t,modP), K.zk) %5 = [1, t, 2*t + 1] ? %[1].mod %6 = t^2 + 3*t + 4 ? K.index %7 = 125

For clarity, we represent elements in the residue
field 𝔽_{5}[t]/(T) as polynomials in the variable t. Whenever the
underlying rational prime does not divide `K.index`

, it is actually
the case that t is the reduction of y in ℚ[y]/(`K.pol`

)
modulo an irreducible factor of `K.pol`

over 𝔽_{p}. In the above
example, 5 divides the index and t is actually the reduction of y/5.

The library syntax is `GEN `

.**nfmodpr**(GEN nf, GEN x, GEN pr)

Transforms the prime ideal *pr* into `modpr`

format necessary
for all operations modulo *pr* in the number field *nf*.
The functions `nfmodpr`

and `nfmodprlift`

allow to project
to and lift from the residue field. The variable v is used to display
finite field elements (see `ffgen`

).

? K = nfinit(y^3-250); ? P = idealprimedec(K, 5)[2]; ? modP = nfmodprinit(K, P, 't); ? K.zk %4 = [1, 1/5*y, 1/25*y^2] ? apply(t->nfmodpr(K,t,modP), K.zk) %5 = [1, t, 2*t + 1] ? %[1].mod %6 = t^2 + 3*t + 4 ? K.index %7 = 125

For clarity, we represent elements in the residue
field 𝔽_{5}[t]/(T) as polynomials in the variable t. Whenever the
underlying rational prime does not divide `K.index`

, it is actually
the case that t is the reduction of y in ℚ[y]/(`K.pol`

)
modulo an irreducible factor of `K.pol`

over 𝔽_{p}. In the above
example, 5 divides the index and t is actually the reduction of y/5.

The library syntax is `GEN `

where **nfmodprinit0**(GEN nf, GEN pr, long v) = -1)`v)`

is a variable number.

Lift the `t_FFELT`

x (from `nfmodpr`

) in the residue field
modulo *pr* to the ring of integers. Vectors and matrices are also
supported. For polynomials, use `apply`

and the present function.

The argument *pr* is either a maximal ideal in `idealprimedec`

format or, preferably, a *modpr* structure from `nfmodprinit`

.
There are no compatibility checks to try and decide whether x is attached
the same residue field as defined by *pr*: the result is undefined
if not.

The function `nfmodpr`

allows to reduce to the residue field.

? K = nfinit(y^3-250); ? P = idealprimedec(K, 5)[2]; ? modP = nfmodprinit(K,P); ? K.zk %4 = [1, 1/5*y, 1/25*y^2] ? apply(t->nfmodpr(K,t,modP), K.zk) %5 = [1, y, 2*y + 1] ? nfmodprlift(K, %, modP) %6 = [1, 1/5*y, 2/5*y + 1] ? nfeltval(K, %[3] - K.zk[3], P) %7 = 1

The library syntax is `GEN `

.**nfmodprlift**(GEN nf, GEN x, GEN pr)

Transforms the number field *nf*
into the corresponding data using current (usually larger) precision. This
function works as expected if *nf* is in fact a *bnf* or a *bnr*
(update structure to current precision). *If* the original
*bnf* structure was *not* computed by `bnfinit(,1)`

, then
this may be quite slow and even fail: many
generators of principal ideals have to be computed and the algorithm may
fail because the accuracy is not sufficient to bootstrap the
required generators and fundamental units.

The library syntax is `GEN `

.
See also **nfnewprec**(GEN nf, long prec)`GEN `

and
**bnfnewprec**(GEN bnf, long prec)`GEN `

.**bnrnewprec**(GEN bnr, long prec)

Given a polynomial T with coefficients in the number field *nf*,
returns the number of real roots of the s(T) where s runs through
the real embeddings of the field specified by optional argument *pl*:

***** *pl* omitted: all r_{1} real places;

***** *pl* an integer between 1 and r_{1}: the embedding attached to
the i-th real root of `nf.pol`

, i.e. `nf.roots[i]`

;

***** *pl* a vector or `t_VECSMALL`

: the embeddings
attached to the *pl*[i]-th real roots of `nf.pol`

.

? nf = nfinit('y^2 - 2); ? nf.sign %2 = [2, 0] ? nf.roots %3 = [-1.414..., 1.414...] ? T = x^2 + 'y; ? nfpolsturm(nf, T, 1) \\ subst(T,y,sqrt(2)) has two real roots %5 = 2 ? nfpolsturm(nf, T, 2) \\ subst(T,y,-sqrt(2)) has no real root %6 = 0 ? nfpolsturm(nf, T) \\ all embeddings together %7 = [2, 0] ? nfpolsturm(nf, T, [2,1]) \\ second then first embedding %8 = [0, 2] ? nfpolsturm(nf, x^3) \\ number of distinct roots ! %9 = [1, 1] ? nfpolsturm(nf, x, 6) \\ there are only 2 real embeddings ! *** at top-level: nfpolsturm(nf,x,6) *** ^ — — — — — -- *** nfpolsturm: domain error in nfpolsturm: index > 2

The library syntax is `GEN `

.**nfpolsturm**(GEN nf, GEN T, GEN pl = NULL)

Let `pol`

be an irreducible integral polynomial defining a number
field K with Galois closure ~{K}. This function is limited to the
Galois groups supported by `nflist`

; in the following ℓ denotes an
odd prime. If Gal(~{K}/ℚ) is D_ℓ, A_{4}, S_{4}, F_{5}
(M_{20}), A_{5}, M_{21} or M_{42},
return a polynomial R defining the corresponding resolvent field (quadratic
for D_ℓ, cyclic cubic for A_{4} and M_{21}, noncyclic cubic for S_{4},
cyclic quartic for F_{5}, A_{5}(6) sextic for A_{5}, and cyclic sextic for
M_{42}). In the A_{5}(6) case, return the A_{5} field of which it is the
resolvent. Otherwise, give a "canonical" subfield, or 0 if the Galois
group is not supported.

The binary digits of *flag* correspond to 0: return a pair [R,f] where f
is a "conductor" whose definition is specific to each group and given
below; 1: return all "canonical" subfields.

Let D be the discriminant of the resolvent field `nfdisc`

(R):

***** In cases C_ℓ, D_ℓ, A_{4}, or S_{4}, disc(K)
= (Df^2)^m with m = (ℓ-1)/2 in the first two cases, and 1 in the last
two.

***** In cases where K is abelian over the resolvent subfield, the conductor
of the relative extension.

***** In case F_{5}, disc(K) = Df^4 if f > 0 or 5^2Df^4 if f < 0.

***** In cases M_{21} or M_{42}, disc(K) = D^mf^6 if f > 0 or
7^3D^mf^6 if f < 0, where m = 2 for M_{21} and m = 1 for M_{42}.

***** In cases A_{5} and A_{5}(6), *flag* is currently ignored.

The library syntax is `GEN `

.**nfresolvent**(GEN pol, long flag)

Roots of the polynomial x in the
number field *nf* given by `nfinit`

without multiplicity (in ℚ
if *nf* is omitted). x has coefficients in the number field (scalar,
polmod, polynomial, column vector). The main variable of *nf* must be
of lower priority than that of x (see Section se:priority). However if the
coefficients of the number field occur explicitly (as polmods) as
coefficients of x, the variable of these polmods *must* be the same as
the main variable of t (see `nffactor`

).

It is possible to input a defining polynomial for *nf*
instead, but this is in general less efficient since parts of an `nf`

structure will then be computed internally. This is useful in two
situations: when you do not need the `nf`

elsewhere, or when you cannot
initialize an `nf`

due to integer factorization difficulties when
attempting to compute the field discriminant and maximal order.

**Caveat.** `nfinit([T, listP])`

allows to compute in polynomial
time a conditional *nf* structure, which sets `nf.zk`

to an order
which is not guaranteed to be maximal at all primes. Always either use
`nfcertify`

first (which may not run in polynomial time) or make sure
to input `nf.pol`

instead of the conditional *nf*: `nfroots`

is
able to recover in polynomial time in this case, instead of potentially
missing a factor.

The library syntax is `GEN `

.
See also **nfroots**(GEN nf = NULL, GEN x)`GEN `

,
corresponding to **nfrootsQ**(GEN x)`nf`

= `NULL`

.

Returns a two-component vector [w,z] where w is the number of roots of
unity in the number field *nf*, and z is a primitive w-th root
of unity. It is possible to input a defining polynomial for *nf*
instead.

? K = nfinit(polcyclo(11)); ? nfrootsof1(K) %2 = [22, [0, 0, 0, 0, 0, -1, 0, 0, 0, 0]~] ? z = nfbasistoalg(K, %[2]) \\ in algebraic form %3 = Mod(-x^5, x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) ? [lift(z^11), lift(z^2)] \\ proves that the order of z is 22 %4 = [-1, -x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1]

This function guesses the number w as the gcd of the #k(v)^{*} for
unramified v above odd primes, then computes the roots in *nf*
of the w-th cyclotomic polynomial. The algorithm is polynomial time with
respect to the field degree and the bitsize of the multiplication table in
*nf* (both of them polynomially bounded in terms of the size of the
discriminant). Fields of degree up to 100 or so should require less than
one minute.

The library syntax is `GEN `

.**nfrootsof1**(GEN nf)

Given a torsion ℤ_{K}-module x attached to the square integral
invertible pseudo-matrix (A,I,J), returns an ideal list
D = [d_{1},...,d_{n}] which is the Smith normal form of x. In other
words, x is isomorphic to ℤ_{K}/d_{1}⨁ ...⨁ ℤ_{K}/d_{n} and d_{i}
divides d_{i-1} for i ≥ 2. If *flag* is nonzero return [D,U,V], where
UAV is the identity.

See Section se:ZKmodules for the definition of integral pseudo-matrix;
briefly, it is input as a 3-component row vector [A,I,J] where
I = [b_{1},...,b_{n}] and J = [a_{1},...,a_{n}] are two ideal lists,
and A is a square n x n matrix with columns (A_{1},...,A_{n}),
seen as elements in K^n (with canonical basis (e_{1},...,e_{n})).
This data defines the ℤ_{K} module x given by
(b_1e_{1}⨁ ...⨁ b_ne_{n}) / (a_1A_{1}⨁ ...⨁ a_nA_{n})
,
The integrality condition is a_{i,j} ∈ b_{i} a_{j}^{-1} for all i,j. If it
is not satisfied, then the d_{i} will not be integral. Note that every
finitely generated torsion module is isomorphic to a module of this form and
even with b_{i} = Z_{K} for all i.

The library syntax is `GEN `

.
Also available:**nfsnf0**(GEN nf, GEN x, long flag)

`GEN `

(**nfsnf**(GEN nf, GEN x)*flag* = 0).

This function is obsolete, use `nfmodpr`

.

Let P be a prime ideal in **modpr** format (see `nfmodprinit`

),
let a be a matrix, invertible over the residue field, and let b be
a column vector or matrix. This function returns a solution of a.x =
b; the coefficients of x are lifted to *nf* elements.

? K = nfinit(y^2+1); ? P = idealprimedec(K, 3)[1]; ? P = nfmodprinit(K, P); ? a = [y+1, y; y, 0]; b = [1, y]~ ? nfsolvemodpr(K, a,b, P) %5 = [1, 2]~

The library syntax is `GEN `

.
This function is normally useless in library mode. Project your
inputs to the residue field using **nfsolvemodpr**(GEN nf, GEN a, GEN b, GEN P)`nfM_to_FqM`

, then work there.

Defining polynomial S over ℚ for the splitting field of
*P* ∈ ℚ[x], that is the smallest field over which P is totally split.
If irreducible, the polynomial P can also be given by a `nf`

structure,
which is more efficient. If d is given, it must be a multiple of the
splitting field degree. Note that if P is reducible the splitting field
degree can be smaller than the degree of P.

If *flag* is non-zero, we assume P to be monic, integral and irreducible and
the return value depends on *flag*:

***** *flag* = 1: return [S,C] where S is as before and C is an
embedding of ℚ[x]/(P) in its splitting field given by a polynomial
(implicitly modulo S, as in `nfisincl`

).

***** *flag* = 2: return [S,C] where C is a rational function whose image
in ℚ[x]/(S) yields an embedding; this avoids inverting the denominator,
which is costly. when the degree of the splitting field is huge.

***** *flag* = 3: return [S, v, p] a data structure allowing to quickly
compute the Galois group of the splitting field, which is used by
`galoissplittinginit`

; more precisely, p is a prime splitting
completely in the splitting field and v is a vector with deg S
elements describing the automorphisms of S acting on the roots
of S modulo p.

? K = nfinit(x^3 - 2); ? nfsplitting(K) %2 = x^6 + 108 ? nfsplitting(x^8 - 2) %3 = x^16 + 272*x^8 + 64 ? S = nfsplitting(x^6 - 8) \\ reducible %4 = x^4 + 2*x^2 + 4 ? lift(nfroots(subst(S,x,a),x^6-8)) %5 = [-a, a, -1/2*a^3 - a, -1/2*a^3, 1/2*a^3, 1/2*a^3 + a] ? P = x^8-2; ? [S,C] = nfsplitting(P,,1) %7 = [x^16 + 272*x^8 + 64, -7/768*x^13 - 239/96*x^5 + 1/2*x] ? subst(P, x, Mod(C,S)) %8 = Mod(0, x^16 + 272*x^8 + 64)

Specifying the degree d of the splitting field can make the computation faster; if d is not a multiple of the true degree, it will be ignored with a warning.

? nfsplitting(x^17-123); time = 3,607 ms. ? poldegree(%) %2 = 272 ? nfsplitting(x^17-123,272); time = 150 ms. ? nfsplitting(x^17-123,273); *** nfsplitting: Warning: ignoring incorrect degree bound 273 time = 3,611 ms.

The complexity of the algorithm is polynomial in the degree d of the
splitting field and the bitsize of T; if d is large the result will
likely be unusable, e.g. `nfinit`

will not be an option:

? nfsplitting(x^6-x-1) [... degree 720 polynomial deleted ...] time = 11,020 ms.

Variant: Also available is
`GEN `

for **nfsplitting**(GEN T, GEN D)*flag* = 0.

The library syntax is `GEN `

.**nfsplitting0**(GEN P, GEN d = NULL, long fl)

Finds all subfields of degree
d of the number field defined by the (monic, integral) polynomial
*pol* (all subfields if d is null or omitted). The result is a vector
of subfields, each being given by [g,h] (default) or simply g (*flag* = 1),
where g is an absolute equation
and h expresses one of the roots of g in terms of the root x of the
polynomial defining *nf*. This routine uses

***** Allombert's `galoissubfields`

when *nf* is Galois (with weakly
supersolvable Galois group).

***** Klüners's or van Hoeij-Klüners-Novocin algorithm
in the general case. The latter runs in polynomial time and is generally
superior unless there exists a small unramified prime p such that *pol*
has few irreducible factors modulo p.

An input of the form `[nf, fa]`

is also allowed, where `fa`

is the
factorisation of *nf.pol* over *nf*, expressed as a famat of
polynomials with coefficients in the variable of `nf`

, in which case the van
Hoeij-Klüners-Novocin algorithm is used.

? pol = x^4 - x^3 - x^2 + x + 1; ? nfsubfields(pol) %2 = [[x, 0], [x^2 - x + 1, x^3 - x^2 + 1], [x^4 - x^3 - x^2 + x + 1, x]] ? nfsubfields(pol,,1) %2 = [x, x^2 - x + 1, x^4 - x^3 - x^2 + x + 1] ? y=varhigher("y"); fa = nffactor(pol,subst(pol,x,y)); ? #nfsubfields([pol,fa]) %5 = 3

The library syntax is `GEN `

.
Also available is **nfsubfields0**(GEN pol, long d, long fl)`GEN `

, corresponding
to **nfsubfields**(GEN nf, long d)*flag* = 0.

Compute the maximal CM subfield of *nf*. Return 0 if *nf* does
not have a CM subfield, otherwise return [g,h] (default) or g (flag = 1)
where g is an absolute
equation and h expresses a root of g in terms of the generator of *nf*.
Moreover, the CM involution is given by X mod g(X) `: — >`

-X mod g(X),
i.e. X mod g(X) is a totally imaginary element.

An input of the form `[nf, fa]`

is also allowed, where `fa`

is the
factorisation of *nf.pol* over *nf*, and *nf* is also allowed to
be a monic defining polynomial for the number field.

? nf = nfinit(x^8 + 20*x^6 + 10*x^4 - 4*x^2 + 9); ? nfsubfieldscm(nf) %2 = [x^4 + 4480*x^2 + 3612672, 3*x^5 + 58*x^3 + 5*x] ? pol = y^16-8*y^14+29*y^12-60*y^10+74*y^8-48*y^6+8*y^4+4*y^2+1; ? fa = nffactor(pol, subst(pol,y,x)); ? nfsubfieldscm([pol,fa]) %5 = [y^8 + ... , ...]

The library syntax is `GEN `

.**nfsubfieldscm**(GEN nf, long fl)

Compute the list of maximal subfields of *nf*. The result is a vector
as in `nfsubfields`

.

An input of the form `[nf, fa]`

is also allowed, where `fa`

is the
factorisation of *nf.pol* over *nf*, and *nf* is also allowed to
be a monic defining polynomial for the number field.

The library syntax is `GEN `

.**nfsubfieldsmax**(GEN nf, long fl)

P and Q being squarefree polynomials in ℤ[X] in the same variable, outputs the simple factors of the étale ℚ-algebra A = ℚ(X, Y) / (P(X), Q(Y)). The factors are given by a list of polynomials R in ℤ[X], attached to the number field ℚ(X)/ (R), and sorted by increasing degree (with respect to lexicographic ordering for factors of equal degrees). Returns an error if one of the polynomials is not squarefree.

Note that it is more efficient to reduce to the case where P and Q are irreducible first. The routine will not perform this for you, since it may be expensive, and the inputs are irreducible in most applications anyway. In this case, there will be a single factor R if and only if the number fields defined by P and Q are linearly disjoint (their intersection is ℚ).

Assuming P is irreducible (of smaller degree than Q for efficiency), it is in general much faster to proceed as follows

nf = nfinit(P); L = nffactor(nf, Q)[,1]; vector(#L, i, rnfequation(nf, L[i]))

to obtain the same result. If you are only interested in the degrees of the
simple factors, the `rnfequation`

instruction can be replaced by a
trivial `poldegree(P) * poldegree(L[i])`

.

The binary digits of *flag* mean

1: outputs a vector of 4-component vectors [R,a,b,k], where R ranges through the list of all possible compositums as above, and a (resp. b) expresses the root of P (resp. Q) as an element of ℚ(X)/(R). Finally, k is a small integer such that b + ka = X modulo R.

2: assume that P and Q define number fields which are linearly disjoint: both polynomials are irreducible and the corresponding number fields have no common subfield besides ℚ. This allows to save a costly factorization over ℚ. In this case return the single simple factor instead of a vector with one element.

A compositum is often defined by a complicated polynomial, which it is
advisable to reduce before further work. Here is an example involving
the field ℚ(ζ_{5}, 5^{1/5}):

```
? L = polcompositum(x^5 - 5, polcyclo(5), 1); \\ list of [R,a,b,k]
? [R, a] = L[1]; \\ pick the single factor, extract R,a (ignore b,k)
? R \\ defines the compositum
%3 = x^20 + 5*x^19 + 15*x^18 + 35*x^17 + 70*x^16 + 141*x^15 + 260*x^14\
+ 355*x^13 + 95*x^12 - 1460*x^11 - 3279*x^10 - 3660*x^9 - 2005*x^8 \
+ 705*x^7 + 9210*x^6 + 13506*x^5 + 7145*x^4 - 2740*x^3 + 1040*x^2 \
- 320*x + 256
? a^5 - 5 \\ a fifth root of 5
%4 = 0
? [T, X] = polredbest(R, 1);
? T \\ simpler defining polynomial for ℚ[x]/(R)
%6 = x^20 + 25*x^10 + 5
? X \\ root of R in ℚ[y]/(T(y))
%7 = Mod(-1/11*x^15 - 1/11*x^14 + 1/22*x^10 - 47/22*x^5 - 29/11*x^4 + 7/22,\
x^20 + 25*x^10 + 5)
? a = subst(a.pol, 'x, X) \\
````a`

in the new coordinates
%8 = Mod(1/11*x^14 + 29/11*x^4, x^20 + 25*x^10 + 5)
? a^5 - 5
%9 = 0

In the above example, x^5-5 and the 5-th cyclotomic polynomial are irreducible over ℚ; they have coprime degrees so define linearly disjoint extensions and we could have started by

? [R,a] = polcompositum(x^5 - 5, polcyclo(5), 3); \\ [R,a,b,k]

The library syntax is `GEN `

.
Also available are
**polcompositum0**(GEN P, GEN Q, long flag)`GEN `

(**compositum**(GEN P, GEN Q)*flag* = 0) and
`GEN `

(**compositum2**(GEN P, GEN Q)*flag* = 1).

Galois group of the nonconstant
polynomial T ∈ ℚ[X]. In the present version **2.16.0**, T must be irreducible
and the degree d of T must be less than or equal to 7. If the
`galdata`

package has been installed, degrees 8, 9, 10 and 11 are also
implemented. By definition, if K = ℚ[x]/(T), this computes the action of
the Galois group of the Galois closure of K on the d distinct roots of
T, up to conjugacy (corresponding to different root orderings).

The output is a 4-component vector [n,s,k,name] with the
following meaning: n is the cardinality of the group, s is its signature
(s = 1 if the group is a subgroup of the alternating group A_{d}, s = -1
otherwise) and name is a character string containing name of the transitive
group according to the GAP 4 transitive groups library by Alexander Hulpke.

k is more arbitrary and the choice made up to version 2.2.3 of PARI is rather
unfortunate: for d > 7, k is the numbering of the group among all
transitive subgroups of S_{d}, as given in "The transitive groups of degree up
to eleven", G. Butler and J. McKay, *Communications in Algebra*, vol. 11,
1983,
pp. 863--911 (group k is denoted T_{k} there). And for d ≤ 7, it was ad
hoc, so as to ensure that a given triple would denote a unique group.
Specifically, for polynomials of degree d ≤ 7, the groups are coded as
follows, using standard notations

In degree 1: S_{1} = [1,1,1].

In degree 2: S_{2} = [2,-1,1].

In degree 3: A_{3} = C_{3} = [3,1,1], S_{3} = [6,-1,1].

In degree 4: C_{4} = [4,-1,1], V_{4} = [4,1,1], D_{4} = [8,-1,1], A_{4} = [12,1,1],
S_{4} = [24,-1,1].

In degree 5: C_{5} = [5,1,1], D_{5} = [10,1,1], M_{20} = [20,-1,1],
A_{5} = [60,1,1], S_{5} = [120,-1,1].

In degree 6: C_{6} = [6,-1,1], S_{3} = [6,-1,2], D_{6} = [12,-1,1], A_{4} = [12,1,1],
G_{18} = [18,-1,1], S_{4}^ -= [24,-1,1], A_{4} x C_{2} = [24,-1,2],
S_{4}^ += [24,1,1], G_{36}^ -= [36,-1,1], G_{36}^ += [36,1,1],
S_{4} x C_{2} = [48,-1,1], A_{5} = PSL_{2}(5) = [60,1,1], G_{72} = [72,-1,1],
S_{5} = PGL_{2}(5) = [120,-1,1], A_{6} = [360,1,1], S_{6} = [720,-1,1].

In degree 7: C_{7} = [7,1,1], D_{7} = [14,-1,1], M_{21} = [21,1,1],
M_{42} = [42,-1,1], PSL_{2}(7) = PSL_{3}(2) = [168,1,1], A_{7} = [2520,1,1],
S_{7} = [5040,-1,1].

This is deprecated and obsolete, but for reasons of backward compatibility,
we cannot change this behavior yet. So you can use the default
`new_galois_format`

to switch to a consistent naming scheme, namely k is
always the standard numbering of the group among all transitive subgroups of
S_{n}. If this default is in effect, the above groups will be coded as:

In degree 1: S_{1} = [1,1,1].

In degree 2: S_{2} = [2,-1,1].

In degree 3: A_{3} = C_{3} = [3,1,1], S_{3} = [6,-1,2].

In degree 4: C_{4} = [4,-1,1], V_{4} = [4,1,2], D_{4} = [8,-1,3], A_{4} = [12,1,4],
S_{4} = [24,-1,5].

In degree 5: C_{5} = [5,1,1], D_{5} = [10,1,2], M_{20} = [20,-1,3],
A_{5} = [60,1,4], S_{5} = [120,-1,5].

In degree 6: C_{6} = [6,-1,1], S_{3} = [6,-1,2], D_{6} = [12,-1,3], A_{4} = [12,1,4],
G_{18} = [18,-1,5], A_{4} x C_{2} = [24,-1,6], S_{4}^ += [24,1,7],
S_{4}^ -= [24,-1,8], G_{36}^ -= [36,-1,9], G_{36}^ += [36,1,10],
S_{4} x C_{2} = [48,-1,11], A_{5} = PSL_{2}(5) = [60,1,12], G_{72} = [72,-1,13],
S_{5} = PGL_{2}(5) = [120,-1,14], A_{6} = [360,1,15], S_{6} = [720,-1,16].

In degree 7: C_{7} = [7,1,1], D_{7} = [14,-1,2], M_{21} = [21,1,3],
M_{42} = [42,-1,4], PSL_{2}(7) = PSL_{3}(2) = [168,1,5], A_{7} = [2520,1,6],
S_{7} = [5040,-1,7].

**Warning.** The method used is that of resolvent polynomials and is
sensitive to the current precision. The precision is updated internally but,
in very rare cases, a wrong result may be returned if the initial precision
was not sufficient.

The library syntax is `GEN `

.
To enable the new format in library mode,
set the global variable **polgalois**(GEN T, long prec)`new_galois_format`

to 1.

This function is *deprecated*, use `polredbest`

instead.
Finds polynomials with reasonably small coefficients defining subfields of
the number field defined by T. One of the polynomials always defines ℚ
(hence has degree 1), and another always defines the same number field
as T if T is irreducible.

All T accepted by `nfinit`

are also allowed here;
in particular, the format `[T, listP]`

is recommended, e.g. with
`listP`

= 10^5 or a vector containing all ramified primes. Otherwise,
the maximal order of ℚ[x]/(T) must be computed.

The following binary digits of *flag* are significant:

1: Possibly use a suborder of the maximal order. The
primes dividing the index of the order chosen are larger than
`primelimit`

or divide integers stored in the `addprimes`

table.
This flag is *deprecated*, the `[T, listP]`

format is more
flexible.

2: gives also elements. The result is a two-column matrix, the first column giving primitive elements defining these subfields, the second giving the corresponding minimal polynomials.

? M = polred(x^4 + 8, 2) %1 = [ 1 x - 1] [ 1/2*x^2 + 1 x^2 - 2*x + 3] [-1/2*x^2 + 1 x^2 - 2*x + 3] [ 1/2*x^2 x^2 + 2] [ 1/4*x^3 x^4 + 2] ? minpoly(Mod(M[2,1], x^4+8)) %2 = x^2 + 2

The library syntax is

(**polred**(GEN T)*flag* = 0). Also available is
`GEN `

(**polred2**(GEN T)*flag* = 2). The function `polred0`

is
deprecated, provided for backward compatibility.

Returns a canonical defining polynomial P for the number field
ℚ[X]/(T) defined by T, such that the sum of the squares of the modulus
of the roots (i.e. the T_{2}-norm) is minimal. Different T defining
isomorphic number fields will yield the same P. All T accepted by
`nfinit`

are also allowed here, e.g. nonmonic polynomials, or pairs
`[T, listP]`

specifying that a nonmaximal order may be used. For
convenience, any number field structure (*nf*, *bnf*,...) can also
be used instead of T.

? polredabs(x^2 + 16) %1 = x^2 + 1 ? K = bnfinit(x^2 + 16); polredabs(K) %2 = x^2 + 1

**Warning 1.** Using a `t_POL`

T requires computing
and fully factoring the discriminant d_{K} of the maximal order which may be
very hard. You can use the format `[T, listP]`

, where `listP`

encodes a list of known coprime divisors of disc(T) (see `??nfbasis`

),
to help the routine, thereby replacing this part of the algorithm by a
polynomial time computation But this may only compute a suborder of the
maximal order, when the divisors are not squarefree or do not include all
primes dividing d_{K}. The routine attempts to certify the result
independently of this order computation as per `nfcertify`

: we try to
prove that the computed order is maximal. If the certification fails,
the routine then fully factors the integers returned by `nfcertify`

.
You can also use `polredbest`

to avoid this factorization step; in this
case, the result is small but no longer canonical.

**Warning 2.** Apart from the factorization of the discriminant of
T, this routine runs in polynomial time for a *fixed* degree.
But the complexity is exponential in the degree: this routine
may be exceedingly slow when the number field has many subfields, hence a
lot of elements of small T_{2}-norm. If you do not need a canonical
polynomial, the function `polredbest`

is in general much faster (it runs
in polynomial time), and tends to return polynomials with smaller
discriminants.

The binary digits of *flag* mean

1: outputs a two-component row vector [P,a], where P is the default
output and `Mod(a, P)`

is a root of the original T.

4: gives *all* polynomials of minimal T_{2} norm; of the two polynomials
P(x) and ± P(-x), only one is given.

16: (OBSOLETE) Possibly use a suborder of the maximal order, *without*
attempting to certify the result as in Warning 1. This makes `polredabs`

behave like `polredbest`

. Just use the latter.

? T = x^16 - 136*x^14 + 6476*x^12 - 141912*x^10 + 1513334*x^8 \ - 7453176*x^6 + 13950764*x^4 - 5596840*x^2 + 46225 ? T1 = polredabs(T); T2 = polredbest(T); ? [ norml2(polroots(T1)), norml2(polroots(T2)) ] %3 = [88.0000000, 120.000000] ? [ sizedigit(poldisc(T1)), sizedigit(poldisc(T2)) ] %4 = [75, 67]

The precise definition of the output of `polredabs`

is as follows.

***** Consider the finite list of characteristic polynomials of primitive
elements of K that are in ℤ_{K} and minimal for the T_{2} norm;
now remove from the list the polynomials whose discriminant do not have
minimal absolute value. Note that this condition is restricted to the
original list of polynomials with minimal T_{2} norm and does not imply that
the defining polynomial for the field with smallest discriminant belongs to
the list !

***** To a polynomial P(x) = x^n +...+ a_{n} ∈ ℝ[x] we attach
the sequence S(P) given by |a_{1}|, a_{1},..., |a_{n}|, a_{n}.
Order the polynomials P by the lexicographic order on the coefficient
vectors S(P). Then the output of `polredabs`

is the smallest
polynomial in the above list for that order. In other words, the monic
polynomial which is lexicographically smallest with respect to the absolute
values of coefficients, favouring negative coefficients to break ties, i.e.
choosing x^3-2 rather than x^3+2.

The library syntax is `GEN `

.
Instead of the above hardcoded numerical flags, one should use an
or-ed combination of**polredabs0**(GEN T, long flag)

***** `nf_PARTIALFACT`

(OBSOLETE): possibly use a suborder of the maximal
order, *without* attempting to certify the result.

***** `nf_ORIG`

: return [P, a], where `Mod(a, P)`

is a root of T.

***** `nf_RAW`

: return [P, b], where `Mod(b, T)`

is a root of P.
The algebraic integer b is the raw result produced by the small vectors
enumeration in the maximal order; P was computed as the characteristic
polynomial of `Mod(b, T)`

. `Mod(a, P)`

as in `nf_ORIG`

is obtained with `modreverse`

.

***** `nf_ADDZK`

: if r is the result produced with some of the above
flags (of the form P or [P,c]), return `[r,zk]`

, where `zk`

is a
ℤ-basis for the maximal order of ℚ[X]/(P).

***** `nf_ALL`

: return a vector of results of the above form, for all
polynomials of minimal T_{2}-norm.

Finds a polynomial with reasonably
small coefficients defining the same number field as T.
All T accepted by `nfinit`

are also allowed here (e.g. nonmonic
polynomials, `nf`

, `bnf`

, `[T,Z_K_basis]`

). Contrary to
`polredabs`

, this routine runs in polynomial time, but it offers no
guarantee as to the minimality of its result.

This routine computes an LLL-reduced basis for an order in ℚ[X]/(T), then
examines small linear combinations of the basis vectors, computing their
characteristic polynomials. It returns the *separable* polynomial P of
smallest discriminant, the one with lexicographically smallest
`abs(Vec(P))`

in case of ties. This is a good candidate for subsequent
number field computations since it guarantees that the denominators of
algebraic integers, when expressed in the power basis, are reasonably small.
With no claim of minimality, though.

It can happen that iterating this functions yields better and better polynomials, until it stabilizes:

? \p5 ? P = X^12+8*X^8-50*X^6+16*X^4-3069*X^2+625; ? poldisc(P)*1. %2 = 1.2622 E55 ? P = polredbest(P); ? poldisc(P)*1. %4 = 2.9012 E51 ? P = polredbest(P); ? poldisc(P)*1. %6 = 8.8704 E44

In this example, the initial polynomial P is the one
returned by `polredabs`

, and the last one is stable.

If *flag* = 1: outputs a two-component row vector [P,a], where P is the
default output and `Mod(a, P)`

is a root of the original T.

? [P,a] = polredbest(x^4 + 8, 1) %1 = [x^4 + 2, Mod(x^3, x^4 + 2)] ? charpoly(a) %2 = x^4 + 8

In particular, the map ℚ[x]/(T) → ℚ[x]/(P),
x`: — >`

`Mod(a,P)`

defines an isomorphism of number fields, which can
be computed as

subst(lift(Q), 'x, a)

if Q is a `t_POLMOD`

modulo T; `b = modreverse(a)`

returns a `t_POLMOD`

giving the inverse of the above map (which should be
useless since ℚ[x]/(P) is a priori a better representation for the number
field and its elements).

The library syntax is `GEN `

.**polredbest**(GEN T, long flag)

This function is obsolete, use polredbest.

The library syntax is `GEN `

.**polredord**(GEN x)

Applies a random Tschirnhausen
transformation to the polynomial x, which is assumed to be nonconstant
and separable, so as to obtain a new equation for the étale algebra
defined by x. This is for instance useful when computing resolvents,
hence is used by the `polgalois`

function.

The library syntax is `GEN `

.**tschirnhaus**(GEN x)

Expresses x on the relative
integral basis. Here, *rnf* is a relative number field extension L/K
as output by `rnfinit`

, and x an element of L in absolute form, i.e.
expressed as a polynomial or polmod with polmod coefficients, *not* on
the relative integral basis.

The library syntax is `GEN `

.**rnfalgtobasis**(GEN rnf, GEN x)

Let K the field represented by
*bnf*, as output by `bnfinit`

. M is a projective ℤ_{K}-module
of rank n (M ⨂ K is an n-dimensional K-vector space), given by a
pseudo-basis of size n. The routine returns either a true ℤ_{K}-basis of
M (of size n) if it exists, or an n+1-element generating set of M if
not.

It is allowed to use a monic irreducible polynomial P in K[X] instead of
M, in which case, M is defined as the ring of integers of K[X]/(P),
viewed as a ℤ_{K}-module.

**Huge discriminants, helping rnfdisc.** The format [T,B] is
also accepted instead of T and computes an order which is maximal at all
maximal ideals specified by B, see `??rnfinit`

: the valuation of D is
then correct at all such maximal ideals but may be incorrect at other primes.

The library syntax is `GEN `

.**rnfbasis**(GEN bnf, GEN M)

Computes the representation of x
as a polmod with polmods coefficients. Here, *rnf* is a relative number
field extension L/K as output by `rnfinit`

, and x an element of
L expressed on the relative integral basis.

The library syntax is `GEN `

.**rnfbasistoalg**(GEN rnf, GEN x)

Characteristic polynomial of
a over *nf*, where a belongs to the algebra defined by T over
*nf*, i.e. *nf*[X]/(T). Returns a polynomial in variable v
(x by default).

? nf = nfinit(y^2+1); ? rnfcharpoly(nf, x^2+y*x+1, x+y) %2 = x^2 + Mod(-y, y^2 + 1)*x + 1

The library syntax is `GEN `

where **rnfcharpoly**(GEN nf, GEN T, GEN a, long var = -1)`var`

is a variable number.

Given a *bnf* structure attached to a number field K, as produced
by `bnfinit`

, and T an irreducible polynomial in K[x]
defining an Abelian extension L = K[x]/(T), computes the class field
theory conductor of this Abelian extension. If T does not define an Abelian
extension over K, the result is undefined; it may be the integer 0 (in
which case the extension is definitely not Abelian) or a wrong result.

The result is a 3-component vector [f,*bnr*,H], where f is the
conductor of the extension given as a 2-component row vector [f_{0},f_ oo ],
*bnr* is the attached `bnr`

structure and H is a matrix in HNF
defining the subgroup of the ray class group on the ray class group generators
`bnr.gen`

; in particular, it is a left divisor of the diagonal matrix
attached to `bnr.cyc`

and |det H |= N = deg T.

***** If *flag* is 1, return [f,*bnrmod*, H], where
`bnrmod`

is now attached to Cl_{f} / Cl_{f}^N, and H is as
before since it contains the N-th powers. This is useful when f contains
a maximal ideal with huge residue field, since the corresponding tough
discrete logarithms are trivialized: in the quotient group, all elements have
small order dividing N. This allows to work in Cl_{f}/H but no
longer in Cl_{f}.

***** If *flag* is 2, only return [f, `fa`

] where `fa`

is the
factorization of the conductor finite part ( = f[1]).

**Huge discriminants, helping rnfdisc.** The format [T,B] is
also accepted instead of T and computes the conductor of the extension
provided it factors completely over the maximal ideals specified by B,
see `??rnfinit`

: the valuation of f_{0} is then correct at all such
maximal ideals but may be incorrect at other primes.

The library syntax is `GEN `

.
Also available is **rnfconductor0**(GEN bnf, GEN T, long flag)`GEN `

when **rnfconductor**(GEN bnf, GEN T)*flag* =
0.

Given a number field K coded by *nf* and a monic
polynomial P ∈ ℤ_{K}[X], irreducible over K and thus defining a relative
extension L of K, applies Dedekind's criterion to the order
ℤ_{K}[X]/(P), at the prime ideal *pr*. It is possible to set *pr*
to a vector of prime ideals (test maximality at all primes in the vector),
or to omit altogether, in which case maximality at *all* primes is tested;
in this situation *flag* is automatically set to 1.

The default historic behavior (*flag* is 0 or omitted and *pr* is a
single prime ideal) is not so useful since
`rnfpseudobasis`

gives more information and is generally not that
much slower. It returns a 3-component vector [*max*, *basis*, v]:

***** *basis* is a pseudo-basis of an enlarged order O produced by
Dedekind's criterion, containing the original order ℤ_{K}[X]/(P)
with index a power of *pr*. Possibly equal to the original order.

***** *max* is a flag equal to 1 if the enlarged order O
could be proven to be *pr*-maximal and to 0 otherwise; it may still be
maximal in the latter case if *pr* is ramified in L,

***** v is the valuation at *pr* of the order discriminant.

If *flag* is nonzero, on the other hand, we just return 1 if the order
ℤ_{K}[X]/(P) is *pr*-maximal (resp. maximal at all relevant primes, as
described above), and 0 if not. This is much faster than the default,
since the enlarged order is not computed.

? nf = nfinit(y^2-3); P = x^3 - 2*y; ? pr3 = idealprimedec(nf,3)[1]; ? rnfdedekind(nf, P, pr3) %3 = [1, [[1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 1, 1]], 8] ? rnfdedekind(nf, P, pr3, 1) %4 = 1

In this example, `pr3`

is the ramified ideal above 3,
and the order generated by the cube roots of y is already
`pr3`

-maximal. The order-discriminant has valuation 8. On the other
hand, the order is not maximal at the prime above 2:

? pr2 = idealprimedec(nf,2)[1]; ? rnfdedekind(nf, P, pr2, 1) %6 = 0 ? rnfdedekind(nf, P, pr2) %7 = [0, [[2, 0, 0; 0, 1, 0; 0, 0, 1], [[1, 0; 0, 1], [1, 0; 0, 1], [1, 1/2; 0, 1/2]]], 2]

The enlarged order is not proven to be `pr2`

-maximal yet. In fact, it
is; it is in fact the maximal order:

? B = rnfpseudobasis(nf, P) %8 = [[1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 1, [1, 1/2; 0, 1/2]], [162, 0; 0, 162], -1] ? idealval(nf,B[3], pr2) %9 = 2

It is possible to use this routine with nonmonic
P = ∑_{i ≤ n} p_{i} X^i ∈ ℤ_{K}[X] if *flag* = 1;
in this case, we test maximality of Dedekind's order generated by
1, p_{n} α, p_{n}α^2 + p_{n-1}α,...,
p_{n}α^{n-1} + p_{n-1}α^{n-2} +...+ p_{1}α.
The routine will fail if P vanishes on the projective line over the residue
field ℤ_{K}/`pr`

(FIXME).

The library syntax is `GEN `

.**rnfdedekind**(GEN nf, GEN pol, GEN pr = NULL, long flag)

Given a pseudo-matrix M over the maximal
order of *nf*, computes its determinant.

The library syntax is `GEN `

.**rnfdet**(GEN nf, GEN M)

Given an *nf* structure attached to a number field K, as output
by `nfinit`

, and a monic irreducible polynomial T ∈ K[x] defining a
relative extension L = K[x]/(T), compute the relative discriminant of L.
This is a vector [D,d], where D is the relative ideal discriminant and
d is the relative discriminant considered as an element of K^{*}/{K^{*}}^2.
The main variable of *nf* *must* be of lower priority than that of
T, see Section se:priority.

**Huge discriminants, helping rnfdisc.** The format [T,B] is
also accepted instead of T and computes an order which is maximal at all
maximal ideals specified by B, see `??rnfinit`

: the valuation of D is
then correct at all such maximal ideals but may be incorrect at other primes.

The library syntax is `GEN `

.**rnfdiscf**(GEN nf, GEN T)

Let *rnf* be a relative
number field extension L/K as output by `rnfinit`

and let x be an
element of L expressed as a polynomial modulo the absolute equation

, or in terms of the absolute ℤ-basis for ℤ*rnf*.pol_{L}
if *rnf* contains one (as in `rnfinit(nf,pol,1)`

, or after
a call to `nfinit(rnf)`

).
Computes x as an element of the relative extension
L/K as a polmod with polmod coefficients.

? K = nfinit(y^2+1); L = rnfinit(K, x^2-y); ? L.polabs %2 = x^4 + 1 ? rnfeltabstorel(L, Mod(x, L.polabs)) %3 = Mod(x, x^2 + Mod(-y, y^2 + 1)) ? rnfeltabstorel(L, 1/3) %4 = 1/3 ? rnfeltabstorel(L, Mod(x, x^2-y)) %5 = Mod(x, x^2 + Mod(-y, y^2 + 1)) ? rnfeltabstorel(L, [0,0,0,1]~) \\ Z_{L}not initialized yet *** at top-level: rnfeltabstorel(L,[0, *** ^ — — — — — — -- *** rnfeltabstorel: incorrect type in rnfeltabstorel, apply nfinit(rnf). ? nfinit(L); \\ initialize now ? rnfeltabstorel(L, [0,0,0,1]~) %6 = Mod(Mod(y, y^2 + 1)*x, x^2 + Mod(-y, y^2 + 1))

The library syntax is `GEN `

.**rnfeltabstorel**(GEN rnf, GEN x)

*rnf* being a relative number
field extension L/K as output by `rnfinit`

and x being an element of
L expressed as a polynomial or polmod with polmod coefficients (or as a
`t_COL`

on `nfinit(rnf).zk`

), computes
x as an element of K as a `t_POLMOD`

if *flag* = 0 and as a `t_COL`

otherwise. If x is not in K, a domain error occurs.

? K = nfinit(y^2+1); L = rnfinit(K, x^2-y); ? L.pol %2 = x^4 + 1 ? rnfeltdown(L, Mod(x^2, L.pol)) %3 = Mod(y, y^2 + 1) ? rnfeltdown(L, Mod(x^2, L.pol), 1) %4 = [0, 1]~ ? rnfeltdown(L, Mod(y, x^2-y)) %5 = Mod(y, y^2 + 1) ? rnfeltdown(L, Mod(y,K.pol)) %6 = Mod(y, y^2 + 1) ? rnfeltdown(L, Mod(x, L.pol)) *** at top-level: rnfeltdown(L,Mod(x,x *** ^ — — — — — — -- *** rnfeltdown: domain error in rnfeltdown: element not in the base field ? rnfeltdown(L, Mod(y, x^2-y), 1) \\ as a t_COL %7 = [0, 1]~ ? rnfeltdown(L, [0,1,0,0]~) \\ not allowed without absolute nf struct *** rnfeltdown: incorrect type in rnfeltdown (t_COL). ? nfinit(L); \\ add absolute nf structure to L ? rnfeltdown(L, [0,1,0,0]~) \\ now OK %8 = Mod(y, y^2 + 1)

If we had started with
`L = rnfinit(K, x^2-y, 1)`

, then the final would have worked directly.

The library syntax is `GEN `

.
Also available is
**rnfeltdown0**(GEN rnf, GEN x, long flag)`GEN `

(**rnfeltdown**(GEN rnf, GEN x)*flag* = 0).

*rnf* being a relative number field extension L/K as output by
`rnfinit`

and x being an element of L, returns the relative norm
N_{L/K}(x) as an element of K.

? K = nfinit(y^2+1); L = rnfinit(K, x^2-y); ? rnfeltnorm(L, Mod(x, L.pol)) %2 = Mod(x, x^2 + Mod(-y, y^2 + 1)) ? rnfeltnorm(L, 2) %3 = 4 ? rnfeltnorm(L, Mod(x, x^2-y))

The library syntax is `GEN `

.**rnfeltnorm**(GEN rnf, GEN x)

*rnf* being a relative
number field extension L/K as output by `rnfinit`

and x being an
element of L expressed as a polynomial or polmod with polmod
coefficients, computes x as an element of the absolute extension L/ℚ as
a polynomial modulo the absolute equation

.*rnf*.pol

? K = nfinit(y^2+1); L = rnfinit(K, x^2-y); ? L.pol %2 = x^4 + 1 ? rnfeltreltoabs(L, Mod(x, L.pol)) %3 = Mod(x, x^4 + 1) ? rnfeltreltoabs(L, Mod(y, x^2-y)) %4 = Mod(x^2, x^4 + 1) ? rnfeltreltoabs(L, Mod(y,K.pol)) %5 = Mod(x^2, x^4 + 1)

The library syntax is `GEN `

.**rnfeltreltoabs**(GEN rnf, GEN x)

*rnf* being a relative number field extension L/K as output by
`rnfinit`

and x being an element of L, returns the relative trace
Tr_{L/K}(x) as an element of K.

? K = nfinit(y^2+1); L = rnfinit(K, x^2-y); ? rnfelttrace(L, Mod(x, L.pol)) %2 = 0 ? rnfelttrace(L, 2) %3 = 4 ? rnfelttrace(L, Mod(x, x^2-y))

The library syntax is `GEN `

.**rnfelttrace**(GEN rnf, GEN x)

*rnf* being a relative number field extension L/K as output by
`rnfinit`

and x being an element of K, computes x as an element of
the absolute extension L/ℚ. As a `t_POLMOD`

modulo

if *rnf*.pol*flag* = 0 and as a `t_COL`

on the absolute field integer basis if
*flag* = 1.

? K = nfinit(y^2+1); L = rnfinit(K, x^2-y); ? L.pol %2 = x^4 + 1 ? rnfeltup(L, Mod(y, K.pol)) %3 = Mod(x^2, x^4 + 1) ? rnfeltup(L, y) %4 = Mod(x^2, x^4 + 1) ? rnfeltup(L, [1,2]~) \\ in terms of K.zk %5 = Mod(2*x^2 + 1, x^4 + 1) ? rnfeltup(L, y, 1) \\ in terms of nfinit(L).zk %6 = [0, 1, 0, 0]~ ? rnfeltup(L, [1,2]~, 1) %7 = [1, 2, 0, 0]~

The library syntax is `GEN `

.**rnfeltup0**(GEN rnf, GEN x, long flag)

Given a number field *nf* as output by `nfinit`

(or simply a monic irreducible integral polynomial defining the field)
and a polynomial *pol* with coefficients in *nf* defining a
relative extension L of *nf*, computes an absolute equation of L
over ℚ.

The main variable of *nf* *must* be of lower priority than that
of *pol* (see Section se:priority). Note that for efficiency, this does
not check whether the relative equation is irreducible over *nf*, but
only if it is squarefree. If it is reducible but squarefree, the result will
be the absolute equation of the étale algebra defined by *pol*. If
*pol* is not squarefree, raise an `e_DOMAIN`

exception.

? rnfequation(y^2+1, x^2 - y) %1 = x^4 + 1 ? T = y^3-2; rnfequation(nfinit(T), (x^3-2)/(x-Mod(y,T))) %2 = x^6 + 108 \\ Galois closure of Q(2^(1/3))

If *flag* is nonzero, outputs a 3-component row vector [z,a,k], where

***** z is the absolute equation of L over ℚ, as in the default
behavior,

***** a expresses as a `t_POLMOD`

modulo z a root α of the
polynomial defining the base field *nf*,

***** k is a small integer such that θ = β+kα
is a root of z, where β is a root of *pol*. It is guaranteed
that k = 0 whenever ℚ(β) = L.

? T = y^3-2; pol = x^2 +x*y + y^2; ? [z,a,k] = rnfequation(T, pol, 1); ? z %3 = x^6 + 108 ? subst(T, y, a) %4 = 0 ? alpha= Mod(y, T); ? beta = Mod(x*Mod(1,T), pol); ? subst(z, x, beta + k*alpha) %7 = 0

The library syntax is `GEN `

.
Also available are
**rnfequation0**(GEN nf, GEN pol, long flag)`GEN `

(**rnfequation**(GEN nf, GEN pol)*flag* = 0) and
`GEN `

(**rnfequation2**(GEN nf, GEN pol)*flag* = 1).

Given a *bnf* attached to a number field K and a projective
ℤ_{K}-module M given by a pseudo-matrix, returns either a true HNF basis
of M if one exists, or zero otherwise. If M is a polynomial with
coefficients in K, replace it by the pseudo-matrix returned by
`rnfpseudobasis`

.

The library syntax is `GEN `

.**rnfhnfbasis**(GEN bnf, GEN M)

Let *rnf* be a relative
number field extension L/K as output by `rnfinit`

and let x be an
ideal of the absolute extension L/ℚ. Returns the relative pseudo-matrix in
HNF giving the ideal x considered as an ideal of the relative extension
L/K, i.e. as a ℤ_{K}-module.

Let `Labs`

be an (absolute) `nf`

structure attached to L,
obtained via `Labs = nfinit(rnf))`

. Then `rnf`

"knows" about
`Labs`

and x may be given in any format
attached to `Labs`

, e.g. a prime ideal or an ideal in HNF wrt.
`Labs.zk`

:

? K = nfinit(y^2+1); rnf = rnfinit(K, x^2-y); Labs = nfinit(rnf); ? m = idealhnf(Labs, 17, x^3+2); \\ some ideal in HNF wrt. Labs.zk ? B = rnfidealabstorel(rnf, m) %3 = [[1, 8; 0, 1], [[17, 4; 0, 1], 1]] \\ pseudo-basis for m as Z_{K}-module ? A = rnfidealreltoabs(rnf, B) %4 = [17, x^2 + 4, x + 8, x^3 + 8*x^2] \\ Z-basis for m in Q[x]/(rnf.polabs) ? mathnf(matalgtobasis(Labs, A)) == m %5 = 1

If on the other hand, we do not have a `Labs`

at hand,
because it would be too expensive to compute, but we nevertheless have
a ℤ-basis for x, then we can use the function with this basis as
argument. The entries of x may be given either modulo `rnf.polabs`

(absolute form, possibly lifted) or modulo `rnf.pol`

(relative form as
`t_POLMOD`

s):

? K = nfinit(y^2+1); rnf = rnfinit(K, x^2-y); ? rnfidealabstorel(rnf, [17, x^2 + 4, x + 8, x^3 + 8*x^2]) %2 = [[1, 8; 0, 1], [[17, 4; 0, 1], 1]] ? rnfidealabstorel(rnf, Mod([17, y + 4, x + 8, y*x + 8*y], x^2-y)) %3 = [[1, 8; 0, 1], [[17, 4; 0, 1], 1]]

The library syntax is `GEN `

.**rnfidealabstorel**(GEN rnf, GEN x)

Let *rnf* be a relative number
field extension L/K as output by `rnfinit`

, and x an ideal of
L, given either in relative form or by a ℤ-basis of elements of L
(see Section se:rnfidealabstorel). This function returns the ideal of K
below x, i.e. the intersection of x with K.

The library syntax is `GEN `

.**rnfidealdown**(GEN rnf, GEN x)

Factor into prime ideal powers the
ideal x in the attached absolute number field L = `nfinit`

(*rnf*).
The output format is similar to the `factor`

function, and the prime
ideals are represented in the form output by the `idealprimedec`

function for L.

? rnf = rnfinit(nfinit(y^2+1), x^2-y+1); ? rnfidealfactor(rnf, y+1) \\ P_{2}^2 %2 = [[2, [0,0,1,0]~, 4, 1, [0,0,0,2;0,0,-2,0;-1,-1,0,0;1,-1,0,0]] 2] ? rnfidealfactor(rnf, x) \\ P_{2}%3 = [[2, [0,0,1,0]~, 4, 1, [0,0,0,2;0,0,-2,0;-1,-1,0,0;1,-1,0,0]] 1] ? L = nfinit(rnf); ? id = idealhnf(L, idealhnf(L, 25, (x+1)^2)); ? idealfactor(L, id) == rnfidealfactor(rnf, id) %6 = 1

Note that ideals of the base field K must be explicitly
lifted to L via `rnfidealup`

before they can be factored.

The library syntax is `GEN `

.**rnfidealfactor**(GEN rnf, GEN x)

*rnf* being a relative number
field extension L/K as output by `rnfinit`

and x being a relative
ideal (which can be, as in the absolute case, of many different types,
including of course elements), computes the HNF pseudo-matrix attached to
x, viewed as a ℤ_{K}-module.

The library syntax is `GEN `

.**rnfidealhnf**(GEN rnf, GEN x)

*rnf* being a relative number
field extension L/K as output by `rnfinit`

and x and y being ideals
of the relative extension L/K given by pseudo-matrices, outputs the ideal
product, again as a relative ideal.

The library syntax is `GEN `

.**rnfidealmul**(GEN rnf, GEN x, GEN y)

Let *rnf* be a relative
number field extension L/K as output by `rnfinit`

and let x be a
relative ideal (which can be, as in the absolute case, of many different
types, including of course elements). This function computes the norm of the
x considered as an ideal of the absolute extension L/ℚ. This is
identical to

idealnorm(rnf, rnfidealnormrel(rnf,x))

but faster.

The library syntax is `GEN `

.**rnfidealnormabs**(GEN rnf, GEN x)

Let *rnf* be a relative
number field extension L/K as output by `rnfinit`

and let x be a
relative ideal (which can be, as in the absolute case, of many different
types, including of course elements). This function computes the relative
norm of x as an ideal of K in HNF.

The library syntax is `GEN `

.**rnfidealnormrel**(GEN rnf, GEN x)

Let *rnf* be a relative number
field extension L/K as output by `rnfinit`

, and *pr* a maximal
ideal of K (*prid*), this function completes the *rnf*
with a *nf* structure attached to L (see Section se:rnfinit)
and returns the vector S of prime ideals of ℤ_{L} above *pr*.

? K = nfinit(y^2+1); rnf = rnfinit(K, x^3+y+1); ? pr = idealprimedec(K, 2)[1]; ? S = rnfidealprimedec(rnf, pr); ? #S %4 = 1

The relative ramification indices and residue degrees
can be obtained as `PR.e / pr.e`

and `PR.f / PR.f`

, if `PR`

is an element of S.

The argument *pr* is also allowed to be a prime number p, in which
case the function returns a pair of vectors `[SK,SL]`

, where `SK`

contains the primes of K above p and `SL`

[i] is the vector of primes
of L above `SK`

[i].

? [SK,SL] = rnfidealprimedec(rnf, 5); ? [#SK, vector(#SL,i,#SL[i])] %6 = [2, [2, 2]]

The library syntax is `GEN `

.**rnfidealprimedec**(GEN rnf, GEN pr)

Let *rnf* be a relative
number field extension L/K as output by `rnfinit`

and let x be a
relative ideal, given as a ℤ_{K}-module by a pseudo matrix [A,I].
This function returns the ideal x as an absolute ideal of L/ℚ.
If *flag* = 0, the result is given by a vector of `t_POLMOD`

s modulo
`rnf.pol`

forming a ℤ-basis; if *flag* = 1, it is given in HNF in terms
of the fixed ℤ-basis for ℤ_{L}, see Section se:rnfinit.

? K = nfinit(y^2+1); rnf = rnfinit(K, x^2-y); ? P = idealprimedec(K,2)[1]; ? P = rnfidealup(rnf, P) %3 = [2, x^2 + 1, 2*x, x^3 + x] ? Prel = rnfidealhnf(rnf, P) %4 = [[1, 0; 0, 1], [[2, 1; 0, 1], [2, 1; 0, 1]]] ? rnfidealreltoabs(rnf,Prel) %5 = [2, x^2 + 1, 2*x, x^3 + x] ? rnfidealreltoabs(rnf,Prel,1) %6 = [2 1 0 0] [0 1 0 0] [0 0 2 1] [0 0 0 1]

The reason why we do not return by default (*flag* = 0) the customary HNF in
terms of a fixed ℤ-basis for ℤ_{L} is precisely because
a *rnf* does not contain such a basis by default. Completing the
structure so that it contains a *nf* structure for L is polynomial
time but costly when the absolute degree is large, thus it is not done by
default. Note that setting *flag* = 1 will complete the *rnf*.

The library syntax is `GEN `

.
Also available is
**rnfidealreltoabs0**(GEN rnf, GEN x, long flag)`GEN `

(**rnfidealreltoabs**(GEN rnf, GEN x)*flag* = 0).

*rnf* being a relative
number field extension L/K as output by `rnfinit`

and x being an
ideal of the relative extension L/K given by a pseudo-matrix, gives a
vector of two generators of x over ℤ_{L} expressed as polmods with polmod
coefficients.

The library syntax is `GEN `

.**rnfidealtwoelement**(GEN rnf, GEN x)

Let *rnf* be a relative number
field extension L/K as output by `rnfinit`

and let x be an ideal of
K. This function returns the ideal xℤ_{L} as an absolute ideal of L/ℚ,
in the form of a ℤ-basis. If *flag* = 0, the result is given by a vector of
polynomials (modulo `rnf.pol`

); if *flag* = 1, it is given in HNF in terms
of the fixed ℤ-basis for ℤ_{L}, see Section se:rnfinit.

? K = nfinit(y^2+1); rnf = rnfinit(K, x^2-y); ? P = idealprimedec(K,2)[1]; ? rnfidealup(rnf, P) %3 = [2, x^2 + 1, 2*x, x^3 + x] ? rnfidealup(rnf, P,1) %4 = [2 1 0 0] [0 1 0 0] [0 0 2 1] [0 0 0 1]

The reason why we do not return by default (*flag* = 0) the customary HNF in
terms of a fixed ℤ-basis for ℤ_{L} is precisely because
a *rnf* does not contain such a basis by default. Completing the
structure so that it contains a *nf* structure for L is polynomial
time but costly when the absolute degree is large, thus it is not done by
default. Note that setting *flag* = 1 will complete the *rnf*.

The library syntax is `GEN `

.
Also available is
**rnfidealup0**(GEN rnf, GEN x, long flag)`GEN `

(**rnfidealup**(GEN rnf, GEN x)*flag* = 0).

Given an *nf* structure attached to a number field K, as output by
`nfinit`

, and a monic irreducible polynomial T in ℤ_{K}[x] defining a
relative extension L = K[x]/(T), this computes data to work in L/K
The main variable of T must be of higher priority
(see Section se:priority) than that of *nf*, and the coefficients of
T must be in K.

The result is a row vector, whose components are technical.
We let m = [K:ℚ] the degree of the base field, n = [L:K] the relative
degree, r_{1} and r_{2} the number of real and complex places of K. Access
to this information via *member functions* is preferred since the
specific data organization specified below will change in the future.

If *flag* = 1, add an *nf* structure attached to L to *rnf*.
This is likely to be very expensive if the absolute degree mn is large,
but fixes an integer basis for ℤ_{L} as a ℤ-module and allows to input
and output elements of L in absolute form: as `t_COL`

for elements,
as `t_MAT`

in HNF for ideals, as `prid`

for prime ideals. Without such
a call, elements of L are represented as `t_POLMOD`

, etc.
Note that a subsequent `nfinit`

(*rnf*) will also explicitly
add such a component, and so will the following functions `rnfidealmul`

,
`rnfidealtwoelt`

, `rnfidealprimedec`

, `rnfidealup`

(with flag 1)
and `rnfidealreltoabs`

(with flag 1). The absolute *nf* structure
attached to L can be recovered using `nfinit(rnf)`

.

*rnf*[1](`rnf.pol`

) contains the relative polynomial T.

*rnf*[2] contains the integer basis [A,d] of K, as
(integral) elements of L/ℚ. More precisely, A is a vector of
polynomial with integer coefficients, d is a denominator, and the integer
basis is given by A/d.

*rnf*[3] (`rnf.disc`

) is a two-component row vector
[𝔡(L/K),s] where 𝔡(L/K) is the relative ideal discriminant
of L/K and s is the discriminant of L/K viewed as an element of
K^{*}/(K^{*})^2, in other words it is the output of `rnfdisc`

.

*rnf*[4](`rnf.index`

) is the ideal index 𝔣, i.e. such
that d(T)ℤ_{K} = 𝔣^2𝔡(L/K).

*rnf*[5](`rnf.p`

) is the list of rational primes dividing the norm
of the relative discriminant ideal.

*rnf*[7] (`rnf.zk`

) is the pseudo-basis (A,I) for the maximal
order ℤ_{L} as a ℤ_{K}-module: A is the relative integral pseudo basis
expressed as polynomials (in the variable of T) with polmod coefficients
in *nf*, and the second component I is the ideal list of the
pseudobasis in HNF.

*rnf*[8] is the inverse matrix of the integral basis matrix, with
coefficients polmods in *nf*.

*rnf*[9] is currently unused.

*rnf*[10] (`rnf.nf`

) is *nf*.

*rnf*[11] is an extension of `rnfequation(K, T, 1)`

. Namely, a
vector [P, a, k, `K.pol`

, T] describing the *absolute*
extension L/ℚ: P is an absolute equation, more conveniently obtained
as `rnf.polabs`

; a expresses the generator α = y mod `K.pol`

of the number field K as an element of L, i.e. a polynomial modulo the
absolute equation P;

k is a small integer such that, if β is an abstract root of T and α the generator of K given above, then P(β + kα) = 0. It is guaranteed that k = 0 if ℚ(β) = L.

**Caveat.** Be careful if k != 0 when dealing simultaneously with
absolute and relative quantities since L = ℚ(β + kα) =
K(α), and the generator chosen for the absolute extension is not the
same as for the relative one. If this happens, one can of course go on
working, but we advise to change the relative polynomial so that its root
becomes β + k α. Typical GP instructions would be

[P,a,k] = rnfequation(K, T, 1); if (k, T = subst(T, x, x - k*Mod(y, K.pol))); L = rnfinit(K, T);

*rnf*[12] is by default unused and set equal to 0. This field is used
to store further information about the field as it becomes available (which
is rarely needed, hence would be too expensive to compute during the initial
`rnfinit`

call).

**Huge discriminants, helping rnfdisc.** When T has a
discriminant which is difficult to factor, it is hard to compute
ℤ_{L}. As in `nfinit`

, the special input format [T,B]
is also accepted, where T is a polynomial as above and B specifies a
list of maximal ideals. The following formats are recognized for B:

***** an integer: the list of all maximal ideals above a rational
prime p < B.

***** a vector of rational primes or prime ideals: the list of all maximal
ideals dividing an element in the list.

Instead of ℤ_{L}, this produces an order which is maximal at all such
maximal ideals primes. The result may actually be a complete and correct
*rnf* structure if the relative ideal discriminant factors completely
over this list of maximal ideals but this is not guaranteed. In general, the
order may not be maximal at primes 𝔭 not in the list such that
𝔭^2 divides the relative ideal discriminant.

The library syntax is `GEN `

.
Also available is
**rnfinit0**(GEN nf, GEN T, long flag)`GEN `

(**rnfinit**(GEN nf,GEN T)*flag* = 0).

T being a relative polynomial with coefficients
in *nf*, return 1 if it defines an abelian extension, and 0 otherwise.

? K = nfinit(y^2 + 23); ? rnfisabelian(K, x^3 - 3*x - y) %2 = 1

The library syntax is `long `

.**rnfisabelian**(GEN nf, GEN T)

Given a *bnf* attached to a number field K and
a projective ℤ_{K}-module M given by a pseudo-matrix, return true (1) if
M is free else return false (0). If M is a polynomial with coefficients
in K, replace it by the pseudo-matrix returned by `rnfpseudobasis`

.

The library syntax is `long `

.**rnfisfree**(GEN bnf, GEN M)

Let *rnf* be a relative number field extension L/K as output
by `rnfinit`

whose degree [L:K] is a power of a prime ℓ.
Return 1 if the ℓ-extension is locally cyclotomic (locally contained in
the cyclotomic ℤ_ℓ-extension of K_{v} at all places v | ℓ), and
0 if not.

? K = nfinit(y^2 + y + 1); ? L = rnfinit(K, x^3 - y); /* = K(zeta_{9}), globally cyclotomic */ ? rnfislocalcyclo(L) %3 = 1 \\ we expect 3-adic continuity by Krasner's lemma ? vector(5, i, rnfislocalcyclo(rnfinit(K, x^3 - y + 3^i))) %5 = [0, 1, 1, 1, 1]

The library syntax is `long `

.**rnfislocalcyclo**(GEN rnf)

Similar to
`bnfisnorm`

but in the relative case. T is as output by
`rnfisnorminit`

applied to the extension L/K. This tries to decide
whether the element a in K is the norm of some x in the extension
L/K.

The output is a vector [x,q], where a = Norm(x)*q. The
algorithm looks for a solution x which is an S-integer, with S a list
of places of K containing at least the ramified primes, the generators of
the class group of L, as well as those primes dividing a. If L/K is
Galois, then this is enough but you may want to add more primes to S to
produce different elements, possibly smaller; otherwise, *flag* is used to
add more primes to S: all the places above the primes p ≤ *flag*
(resp. p|*flag*) if *flag* > 0 (resp. *flag* < 0).

The answer is guaranteed (i.e. a is a norm iff q = 1) if the field is Galois, or, under GRH, if S contains all primes less than 12log^2|disc(M)|, where M is the normal closure of L/K.

If `rnfisnorminit`

has determined (or was told) that L/K is
Galois, and *flag* != 0, a Warning is issued (so that you can set
*flag* = 1 to check whether L/K is known to be Galois, according to T).
Example:

bnf = bnfinit(y^3 + y^2 - 2*y - 1); p = x^2 + Mod(y^2 + 2*y + 1, bnf.pol); T = rnfisnorminit(bnf, p); rnfisnorm(T, 17)

checks whether 17 is a norm in the Galois extension ℚ(β) / ℚ(α), where α^3 + α^2 - 2α - 1 = 0 and β^2 + α^2 + 2α + 1 = 0 (it is).

The library syntax is `GEN `

.**rnfisnorm**(GEN T, GEN a, long flag)

Let K be defined by a root of *pol*, and L/K the extension defined
by the polynomial *polrel*. As usual, *pol* can in fact be an *nf*,
or *bnf*, etc; if *pol* has degree 1 (the base field is ℚ),
polrel is also allowed to be an *nf*, etc. Computes technical data needed
by `rnfisnorm`

to solve norm equations Nx = a, for x in L, and a
in K.

If *flag* = 0, do not care whether L/K is Galois or not.

If *flag* = 1, L/K is assumed to be Galois (unchecked), which speeds up
`rnfisnorm`

.

If *flag* = 2, let the routine determine whether L/K is Galois.

The library syntax is `GEN `

.**rnfisnorminit**(GEN pol, GEN polrel, long flag)

This function is deprecated, use `bnrclassfield`

.

The library syntax is `GEN `

.**rnfkummer**(GEN bnr, GEN subgp = NULL, long prec)

Given a polynomial
*pol* with coefficients in *nf* defining a relative extension L and
a suborder *order* of L (of maximal rank), as output by
`rnfpseudobasis`

(*nf*,*pol*) or similar, gives
[[*neworder*],U], where *neworder* is a reduced order and U is
the unimodular transformation matrix.

The library syntax is `GEN `

.**rnflllgram**(GEN nf, GEN pol, GEN order, long prec)

*bnr* being a big ray
class field as output by `bnrinit`

and *pol* a relative polynomial
defining an Abelian extension, computes the norm group (alias Artin
or Takagi group) corresponding to the Abelian extension of
*bnf* = `bnr.bnf`

defined by *pol*, where the module corresponding to *bnr* is assumed
to be a multiple of the conductor (i.e. *pol* defines a subextension of
bnr). The result is the HNF defining the norm group on the given generators
of `bnr.gen`

. Note that neither the fact that *pol* defines an
Abelian extension nor the fact that the module is a multiple of the conductor
is checked. The result is undefined if the assumption is not correct,
but the function will return the empty matrix `[;]`

if it detects a
problem; it may also not detect the problem and return a wrong result.

The library syntax is `GEN `

.**rnfnormgroup**(GEN bnr, GEN pol)

This function is obsolete: use `rnfpolredbest`

instead.
Relative version of `polred`

. Given a monic polynomial *pol* with
coefficients in *nf*, finds a list of relative polynomials defining some
subfields, hopefully simpler and containing the original field. In the present
version **2.16.0**, this is slower and less efficient than `rnfpolredbest`

.

**Remark.** This function is based on an incomplete reduction
theory of lattices over number fields, implemented by `rnflllgram`

, which
deserves to be improved.

The library syntax is `GEN `

.**rnfpolred**(GEN nf, GEN pol, long prec)

Relative version of `polredabs`

. Given an irreducible monic polynomial
*pol* with coefficients in the maximal order of *nf*, finds a
canonical relative
polynomial defining the same field, hopefully with small coefficients.
Note that the equation is only canonical for a fixed *nf*, using a
different defining polynomial in the *nf* structure will produce a
different relative equation.

The binary digits of *flag* correspond to 1: add information to convert
elements to the new representation, 2: absolute polynomial, instead of
relative, 16: possibly use a suborder of the maximal order. More precisely:

0: default, return P

1: returns [P,a] where P is the default output and a,
a `t_POLMOD`

modulo P, is a root of *pol*.

2: returns *Pabs*, an absolute, instead of a relative, polynomial.
This polynomial is canonical and does not depend on the *nf* structure.
Same as but faster than

polredabs(rnfequation(nf, pol))

3: returns [*Pabs*,a,b], where *Pabs* is an absolute polynomial
as above, a, b are `t_POLMOD`

modulo *Pabs*, roots of `nf.pol`

and *pol* respectively.

16: (OBSOLETE) possibly use a suborder of the maximal order. This makes
`rnfpolredabs`

behave as `rnfpolredbest`

. Just use the latter.

**Warning.** The complexity of `rnfpolredabs`

is exponential in the absolute degree. The function `rnfpolredbest`

runs
in polynomial time, and tends to return polynomials with smaller
discriminants. It also supports polynomials with arbitrary coefficients in
*nf*, neither integral nor necessarily monic.

The library syntax is `GEN `

.**rnfpolredabs**(GEN nf, GEN pol, long flag)

Relative version of `polredbest`

. Given a polynomial *pol*
with coefficients in *nf*, finds a simpler relative polynomial P
defining the same field. As opposed to `rnfpolredabs`

this function does
not return a *smallest* (canonical) polynomial with respect to some
measure, but it does run in polynomial time.

The binary digits of *flag* correspond to 1: add information to convert
elements to the new representation, 2: absolute polynomial, instead of
relative. More precisely:

0: default, return P

1: returns [P,a] where P is the default output and a,
a `t_POLMOD`

modulo P, is a root of *pol*.

2: returns *Pabs*, an absolute, instead of a relative, polynomial.
Same as but faster than

rnfequation(nf, rnfpolredbest(nf,pol))

3: returns [*Pabs*,a,b], where *Pabs* is an absolute polynomial
as above, a, b are `t_POLMOD`

modulo *Pabs*, roots of `nf.pol`

and *pol* respectively.

? K = nfinit(y^3-2); pol = x^2 +x*y + y^2; ? [P, a] = rnfpolredbest(K,pol,1); ? P %3 = x^2 - x + Mod(y - 1, y^3 - 2) ? a %4 = Mod(Mod(2*y^2+3*y+4,y^3-2)*x + Mod(-y^2-2*y-2,y^3-2), x^2 - x + Mod(y-1,y^3-2)) ? subst(K.pol,y,a) %5 = 0 ? [Pabs, a, b] = rnfpolredbest(K,pol,3); ? Pabs %7 = x^6 - 3*x^5 + 5*x^3 - 3*x + 1 ? a %8 = Mod(-x^2+x+1, x^6-3*x^5+5*x^3-3*x+1) ? b %9 = Mod(2*x^5-5*x^4-3*x^3+10*x^2+5*x-5, x^6-3*x^5+5*x^3-3*x+1) ? subst(K.pol,y,a) %10 = 0 ? substvec(pol,[x,y],[a,b]) %11 = 0

The library syntax is `GEN `

.**rnfpolredbest**(GEN nf, GEN pol, long flag)

Given an *nf* structure attached to a number field K, as output by
`nfinit`

, and a monic irreducible polynomial T in ℤ_{K}[x] defining a
relative extension L = K[x]/(T), computes the relative discriminant of L
and a pseudo-basis (A,J) for the maximal order ℤ_{L} viewed as a
ℤ_{K}-module. This is output as a vector [A,J,D,d], where D is the
relative ideal discriminant and d is the relative discriminant considered
as an element of K^{*}/{K^{*}}^2.

? K = nfinit(y^2+1); ? [A,J,D,d] = rnfpseudobasis(K, x^2+y); ? A %3 = [1 0] [0 1] ? J %4 = [1, 1] ? D %5 = [0, -4]~ ? d %6 = [0, -1]~

**Huge discriminants, helping rnfdisc.** The format [T,B] is
also accepted instead of T and produce an order which is maximal at all
prime ideals specified by B, see `??rnfinit`

.

? p = 585403248812100232206609398101; ? q = 711171340236468512951957953369; ? T = x^2 + 3*(p*q)^2; ? [A,J,D,d] = V = rnfpseudobasis(K, T); D time = 22,178 ms. %10 = 3 ? [A,J,D,d] = W = rnfpseudobasis(K, [T,100]); D time = 5 ms. %11 = 3 ? V == W %12 = 1 ? [A,J,D,d] = W = rnfpseudobasis(K, [T, [3]]); D %13 = 3 ? V == W %14 = 1

In this example, the results are identical since D ∩ ℤ factors over primes less than 100 (and in fact, over 3). Had it not been the case, the order would have been guaranteed maximal at primes 𝔭 | p for p ≤ 100 only (resp. 𝔭 | 3). And might have been nonmaximal at any other prime ideal 𝔭 such that 𝔭^2 divided D.

The library syntax is `GEN `

.**rnfpseudobasis**(GEN nf, GEN T)

Given a *nf* attached to a number field K and a projective
module M given by a pseudo-matrix, returns a pseudo-basis (A,I)
(not in HNF in general) such that all the ideals of I except perhaps the
last one are equal to the ring of integers of *nf*. If M is a
polynomial with coefficients in K, replace it by the pseudo-matrix
returned by `rnfpseudobasis`

and return the four-component row vector
[A,I,D,d] where (A,I) are as before and (D,d) are discriminants
as returned by `rnfpseudobasis`

. The ideal class of the last ideal of
I is well defined; it is the Steinitz class of M (its image
in SK_{0}(ℤ_{K})).

The library syntax is `GEN `

.**rnfsteinitz**(GEN nf, GEN M)

Let F be the abelian number field contained in ℚ(ζ_{f})
corresponding to the subgroup H of (ℤ/fℤ)^{*}.
Computes the relative class number h^-(F) = h(F)/h(F^+) of F.
The argument `fH`

encodes F and the data [f,H] as follows:

***** `fH`

= [f, H], where H is given by a vector of
integral generators,

***** `fH`

= [*bnr*, H], where *bnr* is attached to
Cl_{f}(ℚ) and H is a congruence subgroup,

***** `fH`

= [G, H], where G is `idealstar`

(f,1), and H is
a subgroup of (ℤ/fℤ)^ x ,

***** `fH`

= f, where we assume that H = {1}, i.e., F =
ℚ(ζ_{f}),

***** an irreducible integral polynomial defining a primitive element for F.

The algorithm is based on an analytic class number formula:
h^-(F) = Q(F)w(F)∏_{K ⊂ F}N_{ℚ(ζd)/ℚ}
l(-B_{1,χ}/2r) ,
where Q(F) is the unit index of F, w(F) is the number of roots of unity
contained in F and K runs through all imaginary cyclic subfields of F.
For each K, d is the degree [K:ℚ], χ is an arbitrary injective
character of G(K/ℚ) to ℂ^{ x } and the Bernoulli number is given by
B_{1,χ} = (1/f_χ)∑_{a = 1}^{f_χ}aχ(a) = -(1/(2-χ(2))
∑_{1 ≤ a ≤ f_χ/2}χ(a) ,
where f_χ is the conductor of χ, namely the conductor of K.
The unit index Q ∈ {1,2} is difficult to determine in general. If it
could be computed, the function returns [a, b] = [h^-, Q]; else
it returns [2h^-/Q, 0]. More precisely, the second component is 0 unless
we are in one of the following cases:

***** If f = p^a with a prime number p, then Q = 1.

***** If F = ℚ(ζ_{f}), then Q = 1 if and only if f = p^a.

***** If f = 4p^a or p^aq^b with odd prime numbers p,q,
then Q = 1 if and only if [ℚ(ζ_{f}):F] is even.

Finally, the optional parameter p is an *odd* prime number.
If p is given, then `subcyclohminus`

outputs the valuation at p of
h^-(F), in other words the maximal integer e such that
p^e|h^-(F) by evaluating p-adic valuations of Bernoulli numbers.
Since p is odd and Q ∈ {1,2}, the latter can be disregarded and
the result is the same as `valuation(subcyclohminus(f,H)[1], p)`

, but
adding this argument p can be much faster when p does not divide [F:ℚ]
or if a high power of p divides [F:ℚ].

? [a,b] = subcyclohminus(22220); b %1 = 2 \\ = Q ? sizedigit(a) %2 = 4306 \\ huge... ? valuation(a, 101) %3 = 41 ? subcyclohminus(22220, 101) \\ directly compute the valuation %4 = 41

shows that 101^{41} divides h^-(ℚ(ζ_{22220})) exactly.
Let k_{n} be the n-th layer of the cyclotomic ℤ_{3}-extension of
k = ℚ(sqrt{-1501391}); the following computes e_{n} for 1 ≤ n ≤ 3,
where 3^{en} is the 3-part of the relative class number h^-(k_{n}):

? d = 1501391; ? subcyclohminus([9*d, [28,10,8]], 3) %1 = 5 ? subcyclohminus([27*d, [28,188,53]], 3) %2 = 12 ? subcyclohminus([81*d, [161,80,242]], 3) %3 = 26

Note that h^+(k_{n}) is prime to 3 for all n ≥ 0.

The following example computes the 3-part of h^-(F), where F is the subfield of the 7860079-th cyclotomic field with degree 2.3^8.

? p=7860079; a=znprimroot(p)^(2*3^8); ? valuation(subcyclohminus([p,a])[1], 3) time = 1min, 47,896 ms. %2 = 65 ? subcyclohminus([p,a], 3) time = 1,290 ms. %3 = 65

The library syntax is `GEN `

.**subcyclohminus**(GEN fH, GEN p = NULL)

Let F be the abelian number field contained in ℚ(ζ_{f})
corresponding to the subgroup H of (ℤ/fℤ)^{*}, let p > 2 be an odd
prime not dividing [F:ℚ], let F_ oo be the cyclotomic
ℤ_{p}-extension of F and let F_{n} by its n-th layer.
Computes the minus part of Iwasawa polynomials and
λ-invariants attached to F_ oo , using the Stickelberger elements
ξ_{n}^{χ} belonging to F_{n}.

The function is only implemented when p and f are relatively small: both
p^4 and f must fit into an `unsigned long`

integer. The argument
`fH`

encodes the data [f,H] as follows:

***** `fH`

= [f, H], where H is given by a vector of
integral generators,

***** `fH`

= [*bnr*, H], where *bnr* is attached to
Cl_{f}(ℚ) and H is a congruence subgroup,

***** `fH`

= [G, H], where G is `idealstar`

(f,1), and H is
a subgroup of (ℤ/fℤ)^ x ,

***** `fH`

= f, where we assume that H = {1}, i.e., F =
ℚ(ζ_{f}),

***** an irreducible integral polynomial defining a primitive element for F.

If F is quadratic, we also allow p = 2 and more data is output (see below).

For a number field K, we write K_{n} for the n-th layer of the
cyclotomic ℤ_{p}-extension of K. The algorithm considers all cyclic
subfields K of F and all injective odd characters
χ:Gal(K/ℚ)`-->`

ℚ_{p}^{ x }. Let Σ_{n} =
Gal(K_{n}/K), which is cyclic generated by the Frobenius automorphism
σ; we write K_χ = ℚ_{p}(χ),
𝒪_χ = ℤ_{p}[χ] with maximal ideal 𝔭.
The Stickelberger element
ξ_{n}^χ belongs to 𝒪_χ[Σ_{n}];
the polynomial f_{n}^{χ}(x) ∈ 𝒪_{χ}[x]
is constructed from ξ_{n}^{χ} by the correspondence
σ `: — >`

1+x. If n is sufficiently large, then
𝔭 does not divide f_{n}^{χ}(x) and the distinguished polynomial
g_{n}^{χ}(x) ∈ 𝒪_{χ}[x] is uniquely determined by the relation
f_{n}^{χ}(x) = u(x)g_{n}^{χ}(x),u(x) ∈ 𝒪_{χ}[x]^{ x }.
Owing to Iwasawa Main Conjecture proved by Mazur-Wiles, we can define
the Iwasawa polynomial
g_{χ}(x) = lim_{n-->oo }g_{n}^{χ}(x) ∈ 𝒪_{χ}[x].
If r is the smallest integer satisfying
deg g_{n}^{χ} ≤ p^r, then we have
g_{χ}(x) = g_{n}^{χ}(x) (mod 𝔭^{n+1-r}) .
Applying the norm from K_χ down to ℚ_{p}, we obtain polynomials
G_{χ}(x), G_{n}^{χ}(x) ∈ ℤ_{p}[x] satisfying the congruence
G_{χ}(x) = G_{n}^{χ}(x) (mod p^{n+1-r}) .
Note that λ_{p}^-(F) = ∑_{K,χ} deg G_χ(x) is the Iwasawa
λ^--invariant of F, while the μ-invariant μ_{p}(F) is
known to be zero by the theorem of Ferrero-Washington.

If n = 0, the function returns [λ^-_{p}(F)] (the vector may contain
further useful components, see below); for positive n, it returns
all non-constant G_{n}^{χ}(x) mod {p^{n+1-r}} as (K,χ) vary.

? subcycloiwasawa(22220, 41) \\ f = 22220, H = {1} %1 = [217] ? T = polcompositum(x^2 - 42853, polcyclo(5))[1]; ? subcycloiwasawa(T, 5) %3 = [3] ? subcycloiwasawa(T, 5, 4) \\ the sum of the degrees is indeed 3 %4 = [x + 585, x^2 + 405*x]

The first example corresponds to F = ℚ(ζ_{22220}) and shows, that
λ_{41}^-(F) = 217. The second one builds F = ℚ(sqrt{42853},
ζ_{5}) then lists the non-constant G_{4}^χ(x) mod {p^4} for p = 5.
Note that in this case all degrees are ≤ 5 hence r ≤ 1 and
n+1-r ≥ n; so the above also gives G_χ modulo p^4.

We henceforth restrict to the quadratic case, where more information is available, and p = 2 is now allowed: we write F = ℚ(sqrt{d}) of discriminant d ( != 1) and character χ.

**Algorithm and output for n = 0, F = ℚ(sqrt{d.**)}
Currently, only the case d < 0 (F quadratic imaginary,
i.e. χ(-1) = -1) is implemented.

***** If p > 2, the function returns
[λ, κ, [e_{0},...,e_{k}]], where λ = λ_{p}^-(F),
p^{en} denotes the p-part of the class number of F_{n} and e_{n} =
λ n + κ for all n > k. We use Gold's theorem
(Acta Arith. vol.26 (1974), pp. 21--32, vol.26 (1975), pp. 233--240).
Then as soon as e_{n} - e_{n-1} < ϕ(p^n) for some n ≥ 1, we have
λ_{p}(F) = e_{n}-e_{n-1};
if χ(p) = 1 we can weaken the hypothesis to e_{n}-e_{n-1} ≤ ϕ(p^n)
for some n ≥ 1 and obtain the same conclusion.
To compute e_{n} - e_{n-1} we use Bernoulli numbers (`subcyclohminus`

) if
χ(p) = 0 and a much faster algorithm of Gold
(Pacific J. Math. vol.40 (1972), pp.83--88) otherwise.

***** For p = 2, we use Kida's formula (Tohoku Math. J. vol. 31 (1979),
pp. 91--96) and only return [λ^-].

When d > 1, `subcycloiwasawa`

should calculate
λ_{p}(F) = λ_{p}^+(F), which is conjectured to be zero.
But this is not yet implemented.

? subcycloiwasawa(x^2+11111, 2) %1 = [5] /* λ_{2}(ℚ(sqrt{-11111})) = 5 */ ? subcycloiwasawa(x^2+11111, 3) %2 = [1, 0, []] ? subcycloiwasawa(x^2+11111, 11) %3 = [0, 0, []]

This shows that for p = 3, we have λ = 1,
κ = 0, and e_{n} = n for all n ≥ 0.
And at p = 11, we have e_{n} = 0 for all n ≥ 0.

? subcycloiwasawa(x^2+1501391, 3) time = 23 ms. %4 = [14, -16, [2, 5]]

computes e_{n} by Gold's algorithm for F = ℚ(sqrt{-1501391}).
This shows that at p = 3, we have λ = 14, κ = -16, then
e_{0} = 2, e_{1} = 5, and e_{n} = 14n-16 for n ≥ 2.

? subcycloiwasawa(x^2+956238, 3) time = 141 ms. %5 = [14, -19, [1, 3]]

computes e_{n} using Bernoulli numbers for F = ℚ(sqrt{-956238}).
This shows that e_{0} = 1, e_{1} = 3 and e_{n} = 14n-19 for n ≥ 2.

**Algorithm and output for n > 0; F = ℚ(sqrt{d.**)}

***** When d < 0 and n ≥ 1,
`subcycloiwasawa`

computes the Stickelberger element
ξ_{n} = ξ_{n}^{χ} ∈ ℤ_{p}[Σ_{n}] and the Iwasawa polynomial
g(x) = g_{χ}(x) ∈ ℤ_{p}[x]
from the n-th layer F_{n} of the cyclotomic ℤ_{p}-extension of F.
Let q be p (p odd) or 4 (p = 2) and let
q_{0} be the lcm of q and the discriminant d of F, and let q_{n} = q_0p^n.
Then Σ_{n} = Gal(ℚ_{n}/ℚ) = Gal(F_{n}/F) = ⟨s`>`

,
where s is the Frobenius automorphism (ℚ_{n}/ℚ,1+q_{0}) and
ξ_{n} = q_{n}^{-1}∑_{a = 1, (a,qn) = 1}^{qn}aχ(a)^{-1}(ℚ_{n}/ℚ,a)^{-1}
is an element of ℚ[Σ_{n}].
For (p,d) = (2,-1),(2,-2),(2,-3), (2,-6),(3,-3),
we know that λ_{p}(F) = 0 and there is nothing to do.
For the other cases, it is proved that (1/2)ξ_{n} ∈ ℤ_{p}[Σ_{n}].
The polynomial f_{n}(x) ∈ ℤ_{p}[x] is constructed from (1/2)ξ_{n} by the
correspondence s`< — -->`

1+x. If n is sufficiently large, then
p does not divide f_{n}(x) and the distinguished polynomial
g_{n}(x) ∈ ℤ_{p}[x] is uniquely determined by the relation
f_{n}(x) = u(x)g_{n}(x), u(x) ∈ ℤ_{p}[[x]]^{ x }. The Iwasawa polynomial
g(x) is defined by g(x) = lim_{n-->oo }g_{n}(x); if r is the
smallest integer satisfying deg g = λ_{p}(F) ≤ p^r, then we have
g(x) = g_{n}(x) (mod p^{n+1-r}) when p > 2 and modulo 2^{n-r}
otherwise.

Conjecturally, we have further

1. case q_{0} = p: ξ_{n} ∈ ℤ[Σ_{n}].

2. case d = -1 and χ(p) = -1: ξ_{n} ∈ ℤ[Σ_{n}].

3. case d = -3 and χ(p) = -1: (3/2)ξ_{n} ∈ ℤ[Σ_{n}].

4. other cases: (1/2)ξ_{n} ∈ ℤ[Σ_{n}].

Finally, `subcycloiwasawa`

outputs [g] where
g is g_{n}(x) mod {p^{n+1-r}} (p odd) or mod {2^{n-r}} (p = 2).

? subcycloiwasawa(x^2+239, 3, 10) %6 = [x^6 + 18780*x^5 + 14526*x^4 + 18168*x^3 + 3951*x^2 + 1128*x]

This is g(x) mod {3^9}. Indeed, n = 10, λ = 6 (the degree), hence r = 2 and n + 1 - r = 2.

***** When d > 1 and n ≥ 1, ξ_{n}^{*} ∈ ℚ[Σ_{n}] is constructed from
χ^ *= χ^{-1}ω, where χ is the character of F = ℚ(sqrt{d})
and ω is the Teichmüller character mod q. Next we construct
f_{n}^{*}(x) ∈ ℤ_{p}[x] from (1/2)ξ_{n}^{*} by the correspondence
s^{-1}`< — -->`

(1+x)(1+q_{0})^{-1} and define the distinguished
polynomial g_{n}^{*}(x) ∈ ℤ_{p}[x] using f_{n}^{*}(x).
Then g^{*}(x) = lim_{n-->oo }g_{n}^{*}(x) is the Iwasawa polynomial,
which has a connection with Greenberg conjecture for F.
Let r be the smallest integer satisfying deg g^{*} ≤ p^r,
then we have g^{*}(x) = g^{*}_{n}(x) (mod p^{n+1-r})
when p > 2 and g^{*}(x) = g^{*}_{n}(x) (mod 2^{n-r}) when p = 2.
Finally, `subcycloiwasawa`

outputs [g^{*}] where
g^{*} is g_{n}^{*}(x) mod {p^{n+1-r}} (p odd) or mod {2^{n-r}} (p = 2).

? subcycloiwasawa(x^2-13841, 2, 19) time = 1min, 17,238 ms. %7 = [x^3 + 30644*x^2 + 126772*x + 44128]

This is g^{*}(x) mod {2^{17}} (r = 2), the distinguished polynomial
treated in a paper of T. Fukuda, K. Komatsu, M. Ozaki and T. Tsuji
(Funct. Approx. Comment. Math. vol.54.1, pp. 7--17, 2016).

The library syntax is `GEN `

.**subcycloiwasawa**(GEN fH, GEN p, long n)

Let F be the abelian number field contained in ℚ(ζ_{f})
corresponding to the subgroup H of (ℤ/fℤ)^{*}, let p > 2 be an odd
prime not dividing [F:ℚ]. Computes the p-Sylow subgroup A_{F} of the
ideal class group using an unconditional algorithm of M. Aoki and T. Fukuda
(LNCS. vol.4076, pp.56--71, 2006).

The argument `fH`

encodes the data [f,H] as follows:

***** `fH`

= [f, H], where H is given by a vector of
integral generators,

***** `fH`

= [*bnr*, H], where *bnr* is attached to
Cl_{f}(ℚ) and H is a congruence subgroup,

***** `fH`

= [G, H], where G is `idealstar`

(f,1), and H is
a subgroup of (ℤ/fℤ)^ x ,

***** `fH`

= f, where we assume that H = {1}, i.e., F =
ℚ(ζ_{f}),

***** an irreducible integral polynomial defining a primitive element for
F.

The result is a 6-component vector v, and components 2 or 3
can be left empty or only partially computed to save time (see *flag* below):

v[1] is p.

v[2] contains [E, [e_{1},...,e_{k}]] with E = ∑_{i} e_{i}, meaning that
the order of A_{F}^+ is p^E and its cyclic structure is ℤ/p^{e1}ℤ x
...ℤ/p^{ek}ℤ

v[3] similarly describes the order and the structure of A_{F}^-.

v[4] contains the structure of Gal(F/ℚ) as a product of cyclic groups (elementary divisors).

v[5] is the number of cyclic subfields K of F except for ℚ.

v[6] is the number of ℚ_{p}-conjugacy classes of injective
characters χ:Gal(K/ℚ)`-->`

ℚ_{p}^{ x }.

A vector of primes p is also accepted and the result is then a vector of vectors as above, in the same order as the primes.

The group A_{F} is the direct sum of A_{F}^+ and A_{F}^-; each of A_{F}^+
and A_{F}^- is decomposed into χ-parts A_χ.
By default, the function computes only |A_{F}^-| and an upper bound
for |A_{F}^+| (expected to be equal to |A_{F}^+|) separately with
different algorithms. This is expected to be fast. The behavior is controled
by the binary digits of *flag*:

1: if |A_{F}^+| or |A_{F}^-| is computed, also determines its group structure
and guarantees informations about A_{F}^+. This last part is usually
costly.

2: do not compute quantities related to A_{F}^+ (the corresponding (e_{i})
in v[2] is replaced with a dummy empty vector).

4: do not compute quantities related to A_{F}^- (the corresponding (e_{i})
in v[3] is replaced with a dummy empty vector).

8: ignores proper subfields of F. This is motivated by the following kind
of problems: let ℚ(p^k) be the k-th layer of the cyclotomic
ℤ_{p}-extenson of ℚ and define ℚ(n) = ℚ(p_{1}^{e1})...ℚ(p_{r}^{er})
when n factors as n = p_{1}^{e1}...p_{r}^{er}, which is a real cyclic
field of degree n satisfying ℚ(n) ⊂ ℚ(m) when n | m. What are
the prime factors of the class number h(n) of ℚ(n) ? The new prime
factors of h(n), not occurring in a lower level, will all be present
when using this *flag*.

The other value are technical and only useful when bit 1 (certification and structure) is set; do not set them unless you run into difficulties with default parameters.

16: when this bit is set, the function tries to save memory, sacrificing speed; this typically uses half the memory for a slowdown of a factor 2.

32: likely to speed up the algorithm when the rank of A_χ is large and
to create a minor slowdown otherwise. Though the effect is restricted, the
3-class group of ℚ(sqrt{15338}, ζ_{5}) is computed 4 times faster
when this bit is set (see below).

**Examples.** With default *flag* = 0, the function (quickly)
determines the exact value of |A_{F}^-| and a rigorous upper bound of |A_{F}^+|
which is expected to be equal to |A_{F}^+|; of course, when the upper bound
is 0, we know for sure that A_{F}^+ is trivial. With *flag* = 1 we obtain the
group structure of A_{F} completely and guarantee the informations about
A_{F}^+ (slow).

? subcyclopclgp(22220, 101) time = 113 ms. %1 = [101, [0, []], [41, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]], [100, 20, 2, 2], 479, 7999]

This computes the 101-part A_{F} of the ideal class group of
F = ℚ(ζ_{22220}).
The output says that A_{F}^ += 0, which is rigorous (since trivial),
and |A_{F}^- |= 101^{41}, more precisely A_{F}^- is isomorphic to
(ℤ/101ℤ)^{41} which is also rigorous
(since the description of A_{F}^- is always rigorous). The Galois group
Gal(F/ℚ) is ℤ/100ℤ⨁ ℤ/20ℤ⨁ ℤ/2ℤ⨁ ℤ/2ℤ.
The field F has 479 cyclic subfields different from ℚ and
there are 7999 ℚ_{101}-conjugacy classes of injective characters
χ:Gal(K/ℚ)`-->`

ℚ_{101}^{ x }.

? subcyclopclgp(22220, 11) time = 83 ms. %2 = [11, [2, [1, 1]], [16, []], [100, 20, 2, 2], 479, 1799]

This computes the 11-part A_{F} for the same F. The result says that
|A_{F}^+ |= 11^2, A^+_{F} is isomorphic to (ℤ/11ℤ)^2 which is not rigorous
and is only an upper bound, and |A_{F}^- |= 11^{16} which is rigorous. The
group structure of A_{F}^- is unknown.

? subcyclopclgp(22220, 11, 1) time = 185 ms. %3 = [11, [2, [1, 1]], [16, [2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]], [100, 20, 2, 2], 479, 1799]

now guarantees that A^+_{F} is isomorphic to (ℤ/11ℤ)^2
and determines that A_{F}^- is isomorphic to ℤ/11^2ℤ⨁ (ℤ/11ℤ)^{14},
at the expense of slightly increasing the running time.

We now try a much harder example: F = ℚ(sqrt{36322},ζ_{5}), which
we could define using f = 726440 and H = [41, 61, 111, 131] (prove it!).
We will use a defining polynomial instead:

? T = polcompositum(x^2-36322, polcyclo(5), 2); ? subcyclopclgp(T, 5) \\ fast when non rigorous for A^+ time = 82 ms. %4 = [5, [1, [1]], [4, []], [4, 2], 5, 7] \\ try to certify; requires about 2GB of memory ? subcyclopclgp(T, 5, 1) *** subcyclopclgp: the PARI stack overflows ! current stack size: 1000003072 (1907.352 Mbytes) ? default(parisizemax,"2G"); ? subcyclopclgp(T, 5, 1) \\ with more memory, we get an answer time = 36,201 ms. %6 = [5, [1, [1]], [4, [3, 1]], [4, 2], 5, 7] \\ trying to reduce memory use does not work (still need 2GB); slower ? subcyclopclgp(T, 5, 1+16) time = 39,450 ms.

This shows that A_{F}^+ is isomorphic to ℤ/5ℤ and
A_{F}^- is isomorphic to ℤ/5^3ℤ⨁ ℤ/5ℤ for p = 5. For this example,
trying to reduce memory use with *flag* = 1+16 fails: the computation becomes
slower and still needs 2GB; *flag* = 1+16+32 is a disaster: it requires about
8GB and 9 minutes of computation.

Here's a situation where the technical flags make a difference:
let F = ℚ(sqrt{15338}, ζ_{5}).

? T = polcompositum(x^2-15338, polcyclo(5), 2); ? subcyclopclgp(T, 3) time = 123 ms. %2 = [3, [1, [1]], [4, []], [4, 2], 5, 5] ? subcyclopclgp(T, 3, 1) \\ requires a stack of 8GB time = 4min, 47,822 ms. %3 = [3, [1, [1]], [4, [1, 1, 1, 1]], [4, 2], 5, 5] ? subcyclopclgp(T, 3, 1+16); time = 7min, 20,876 ms. \\ works with 5GB, but slower ? subcyclopclgp(T, 3, 1+32); time = 1min, 11,424 ms. \\ also works with 5GB, 4 times faster than original ? subcyclopclgp(T, 3, 1+16+32); time = 1min, 47,285 ms. \\ now works with 2.5GB

Let F = ℚ(106) defined as above; namely, F is the composite field
of ℚ(sqrt{2}) and the subfield of ℚ(ζ_{53^2}) with degree 53.
This time we shall build the compositum using class field theory:

? Q = bnfinit(y); ? bnr1 = bnrinit(Q, 8); H1 = Mat(2); ? bnr2 = bnrinit(Q, [53^2, [1]]); H2 = Mat(53); ? [bnr,H] = bnrcompositum([bnr1, H1], [bnr2, H2]); ? subcyclopclgp([bnr,H], 107) time = 10 ms. %5 = [107, [1, [1]], [0, []], [106], 3, 105] ? subcyclopclgp([bnr,H], 107, 1) \\ requires 2.5GB time = 15min, 13,537 ms. %6 = [107, [1, [1]], [0, []], [106], 3, 105]

Both results are identical (and they were expected to be), but only the second is rigorous. Flag bit 32 has a minor impact in this case (reduces timings by 20 s.)

The library syntax is `GEN `

.**subcyclopclgp**(GEN fH, GEN p, long flag)

*cyc* being a vector of positive integers giving the cyclic
components for a finite Abelian group G (or any object which has a
`.cyc`

method), outputs the list of subgroups of G. Subgroups are
given as HNF left divisors of the SNF matrix corresponding to G.

If *flag* = 0 (default) and *cyc* is a *bnr* structure output by
`bnrinit`

, gives only the subgroups whose modulus is the conductor.
Otherwise, all subgroups are given.

If *bound* is present, and is a positive integer, restrict the output to
subgroups of index less than *bound*. If *bound* is a vector
containing a single positive integer B, then only subgroups of index
exactly equal to B are computed. For instance

? subgrouplist([6,2]) %1 = [[6, 0; 0, 2], [2, 0; 0, 2], [6, 3; 0, 1], [2, 1; 0, 1], [3, 0; 0, 2], [1, 0; 0, 2], [6, 0; 0, 1], [2, 0; 0, 1], [3, 0; 0, 1], [1, 0; 0, 1]] ? subgrouplist([6,2],3) \\ index less than 3 %2 = [[2, 1; 0, 1], [1, 0; 0, 2], [2, 0; 0, 1], [3, 0; 0, 1], [1, 0; 0, 1]] ? subgrouplist([6,2],[3]) \\ index 3 %3 = [[3, 0; 0, 1]] ? bnr = bnrinit(bnfinit(x), [120,[1]], 1); ? L = subgrouplist(bnr, [8]);

In the last example, L corresponds to the 24 subfields of
ℚ(ζ_{120}), of degree 8 and conductor 120 oo (by setting *flag*,
we see there are a total of 43 subgroups of degree 8).

? vector(#L, i, galoissubcyclo(bnr, L[i]))

will produce their equations. (For a general base field, you would
have to rely on `bnrstark`

, or `bnrclassfield`

.)

**Warning.** This function requires factoring the exponent of G.
If you are only interested in subgroups of index n (or dividing n), you
may considerably speed up the function by computing the subgroups of
G/G^n, whose cyclic components are `apply(x- > gcd(n,x), C)`

(where
C gives the cyclic components of G). If you want the *bnr* variant,
now is a good time to use `bnrinit(,,, n)`

as well, to directly compute
the ray class group modulo n-th powers.

The library syntax is `GEN `

.**subgrouplist0**(GEN cyc, GEN bound = NULL, long flag)