Local computation of differents and discriminants

Abstract

We obtain several results on the computation of different and discriminant ideals of finite extensions of local fields. As an application, we deduce routines to compute the p-adic valuation of the discriminant Disc(f ), and the resultant Res(f, g), for polynomials f (x), g(x) ∈ A[x], where A is a Dedekind domain and p is a non-zero prime ideal of A with finite residue field. These routines do not require the computation of neither Disc(f ) nor Res(f, g); hence, they are useful in cases where this latter computation is inefficient because the polynomials have a large degree or very large coefficients.

Residual ideals of MacLane valuations

“…as shown in Proposition 1.12, Corollary 5.5, and equation (9), respectively. The index, the exponent and the conductor of a prime polynomial are also Okutsu invariants admitting explicit formulas in terms of the basic invariants e i , f i , h i [13].…”

“…These operators make the whole theory constructive and well-suited to computational applications. These ideas led to the design of several fast algorithms to perform arithmetic tasks in global fields [2,4,6,7,9,13].…”

“…This result reveals that MacLane's original approach is best-suited for computational applications, because the elements in the set M may be described in terms of discrete parameters which are easily manipulated by a computer. As a consequence, all algorithmic developments based on the Montes algorithm [4,6,7,9,13] admit a more elegant description and a natural extension to arbitrary discrete valued fields. However, we postpone the discussion of these computational aspects to a forthcoming paper [8].…”