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 $\mathfrak {p}$-adic valuation of the discriminant $\operatorname {Disc}(f)$, and the resultant $\operatorname {Res}(f,g)$, for polynomials $f(x),g(x)\in A[x]$, where $A$ is a Dedekind domain and $\mathfrak {p}$ is a non-zero prime ideal of $A$ with finite residue field. These routines do not require the computation of either $\operatorname {Disc} (f)$ or $\operatorname {Res}(f,g)$; hence, they are useful in cases where this latter computation is inefficient because the polynomials have a large degree or very large coefficients.