\l TAN.log

f = x^4 - 2*x^3 + x^2 - 5;
polisirreducible(f)

g = polcyclo(30)

Mod(x,f)^5

lift(Mod(x,g)^15)

polcompositum(x^2-2,x^2-3)

polcompositum(x^4-2, x^4-2*x^2-1)
    x^8 - 4*x^6 + 22*x^4 - 36*x^2 + 49]

L = polcompositum(x^3-2, x^3-2)

nfrootsof1(L[2])

nfisisom(x^2-5, x^2-2)
nfisisom(x^2-5, x^2-x-1)

nfisincl(x^2-5,polcyclo(5))
nfisincl(x^2+5,polcyclo(5))

nfsubfields(x^6 + 40*x^3 + 1372,,1)
    x^3 - 2, x^6 + 40*x^3 + 1372]

nfsubfields(x^8 + 3*x^4 + 9, 2)
    [x^2 - 12, -2/3*x^6],
    [x^2 + 12*x + 144, 4*x^4]]

nfsubfieldsmax(x^8-4*x^5+7*x^4-x^2+x+1, 1)

{h = x^5 + 7*x^4 + 22550*x^3 - 281686*x^2 
  - 85911*x + 3821551};
polredbest(h)

K = nfinit(f);

K.pol
K.sign

K.disc
K.zk
w = K.zk[2];

z = varhigher("z");
nffactor(K, subst(K.pol,x,z))
[           z + Mod(-x, x^4 - 2*x^3 + x^2 - 5) 1]
[        z + Mod(x - 1, x^4 - 2*x^3 + x^2 - 5) 1]
[z^2 - z + Mod(x^2 - x, x^4 - 2*x^3 + x^2 - 5) 1]

lift(nfroots(K, subst(K.pol,x,z)))

nfalgtobasis(K,x^2)

nfbasistoalg(K,[1,1,1,1]~)

nfeltmul(K,[1,-1,0,0]~,x^2)

nfeltnorm(K,x-2)
nfelttrace(K,[0,1,2,0]~)

charpoly(nfbasistoalg(K,[1,2,0,0]~))
minpoly(nfbasistoalg(K,[1,2,0,0]~))

nfeltembed(K,x^3+x)
    11.033224646287677151457919656132410589,
    -2.3541019662496845446137605030969143532
    - 0.33268570002014959478470322160341519810*I]

dec = idealprimedec(K,5);
#dec
[pr1,pr2] = dec;

pr1.f
pr1.e

pr1.gen

pr2.f
pr2.e

p11 = idealprimedec(K,11)[1]; p11.f
modpr = nfmodprinit(K,p11,v);
a = nfmodpr(K,x^2+x+1,modpr)
a^11
a^121

nfmodprlift(K,a,modpr)

idealhnf(K,pr1)
[5 3 4 3]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]

a = idealhnf(K,[23, 10, -5, 1]~)
[260   0 228 123]
[  0 260 123 105]
[  0   0   1   0]
[  0   0   0   1]

idealnorm(K,a)

idealpow(K,pr2,3)
[25 15 21 7]
[ 0  5  2 4]
[ 0  0  1 0]
[ 0  0  0 1]
idealnorm(K,idealadd(K,a,pr2))

[n,b] = idealtwoelt(K,a)
idealadd(K,n,b) == a

idealval(K,a,pr2)

idealismaximal(K,a)
idealismaximal(K,idealhnf(K,pr1)) != 0

fa = idealfactor(K,a);
#fa[,1]

[fa[1,1].p, fa[1,1].f, fa[1,1].e, fa[1,2]]

[fa[2,1].p, fa[2,1].f, fa[2,1].e, fa[2,2]]
fa[2,1]==pr1

[fa[3,1].p, fa[3,1].f, fa[3,1].e, fa[3,2]]

b = idealchinese(K,[pr1,2;pr2,1],[1,-1]);

nfeltval(K,b-1,pr1)
nfeltval(K,b+1,pr2)

nfeltsign(K,b)

c = idealchinese(K,[[pr1,2;pr2,1],[1,1]],[1,-1]);
nfeltsign(K,c)

K2 = bnfinit(K);
K2.nf == K
K2.no

K2.reg

lift(K2.tu)
K2.tu[1]==nfrootsof1(K)[1]

lift(K2.fu)

L = bnfinit(x^3 - x^2 - 54*x + 169);
L.cyc

L.gen

pr = idealprimedec(L,13)[1]
[dl,g] = bnfisprincipal(L,pr);
dl

g
{idealhnf(L,pr) == idealmul(L,g,
    idealfactorback(L,L.gen,dl))}

[dl2,g2] = bnfisprincipal(L,idealpow(L,pr,2));
dl2

g2
idealhnf(L,g2) == idealpow(L,pr,2)

u = [0,2,1]~;
nfeltnorm(L,u)

bnfisunit(L,u)
lift(L.fu)
lift(L.tu)

M = bnfinit(x^2-3019);
M.fu

M = bnfinit(x^2-3019,1);
lift(M.fu)
  871236169789*x - 117525625416599410184425264152
  37539460392094825860314330]

D = 10^9 + 1273;
N = bnfinit(x^2-D,1);
bnfunits(N)
N.fu

bnfisprincipal(N,P)
  for generators, not given.
bnfisprincipal(N,P,4)
bnfisprincipal(N,P,3)

P = idealprimedec(N,2)[1];
S = idealprimedec(N,2);
U = bnfunits(N,S);
#U[1]

bnfisunit(N,2)
bnfisunit(N,2,U)