Kurt Foster on Fri, 06 Mar 2009 16:55:06 +0100

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Finding the "norm" of a relative polynomial

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.

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?