|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 = 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 ? Cheers, Bill.