Bill Allombert on Fri, 06 Mar 2009 17:58:00 +0100

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Re: Finding the "norm" of a relative polynomial

On Fri, Mar 06, 2009 at 08:52:57AM -0700, Kurt Foster wrote:
> Suppose I have K = nfinit(f) where f is monic and irreducible in Z[y]  
> (variable of lower priority than x) and R is the ring of algebraic  
> integers in K.  Let L = rnfinit(K, T) where T is a monic irreducible  
> polynomial in R[x].  Then (according to tha manual users.pdf)
> L[11] = rnfequation(K,T,1) = [P, a, k],
> where P is a defining polynomial for L/Q, a is a polynomial in Q[x] of  
> degree less than poldegree(P) such that
> Mod(f(a), P) = 0 [that is, a is a zero of f expressed as a polynomial  
> in x with rational coefficients, where x is a zero of P],
> and k is an integer such that if T(beta) = f(alpha) = 0, then P(beta +  
> k*alpha) = 0.
> Now if k = 0, P is the obvious "norm" of T from R[x] to Z[x].  But if  
> k != 0, it isn't.  And I don't know how to predict when k != 0.

k!=0 iff P for k=0 is not squarefree iff P for k=0 is reducible.

> Now, because of some special algebraic properties of the relative  
> polynomial T, (and not connected with the information given by  
> rnfinit()), I want that norm polynomial.  Now if K/Q is a Galois  
> extension, the "Galois polynomials" for conjugating the zeroes of f in  
> Z[y] give explicit expressions for the algebraic conjugates of the  
> coefficients of T in R[x], so I have something that's not too  
> horrendous that is guaranteed to give me what I want.
> But if K/Q is *not* Galois, I'm not sure what to do.  I *could* try  
> using numerical approximations of the zeroes of f, multiplying  
> approximate conjugates of T, and rounding, but I'd need to know the  
> approximations were good enough to give the correct answer.  Or, I  
> could use resultants to get a polynomial of degree [L:Q]*[K:Q] whose  
> factors included the polynomial I want.  But this seems rather  
> cumbersome.  Is there a quicker and slicker method to get the "norm"  
> polynomial wen k != 0?

Why not use a relative resultant ?