On the computation of all extensions of a 𝑝-adic field of a given degree

Abstract

Let k \mathbf {k} be a p p -adic field. It is well-known that k \mathbf {k} has only finitely many extensions of a given finite degree. Krasner has given formulae for the number of extensions of a given degree and discriminant. Following his work, we present an algorithm for the computation of generating polynomials for all extensions K / k \mathbf {K}/\mathbf {k} of a given degree and discriminant.