  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 $\dsc(f)$, and
the resultant $\res(f,g)$, for polynomials $f(x),g(x)\in A[x]$, 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 $\dsc(f)$ 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.
