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 \mathfrak {p} -adic valuation of the discriminant Disc ⁡ ( f ) \operatorname {Disc}(f) , and the resultant Res ⁡ ( f , g ) \operatorname {Res}(f,g) , for polynomials f ( x ) , g ( x ) ∈ A [ x ] f(x),g(x)\in A[x] , where A A is a Dedekind domain and p \mathfrak {p} is a non-zero prime ideal of A A with finite residue field. These routines do not require the computation of either Disc ⁡ ( f ) \operatorname {Disc} (f) or Res ⁡ ( f , g ) \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.