Determination of the number of isomorphism classes of extensions of a $\kp$-adic field

TL;DR

This paper provides a formula to count the isomorphism classes of extensions of a p-adic field based on ramification and inertia, using group theory, Krasner's formula, and class field theory.

Contribution

It introduces a new explicit formula for enumerating extension classes of p-adic fields depending only on ramification and inertia.

Findings

Number of classes depends only on ramification and inertia.

Formula derived from group theory, Krasner's formula, and class field theory.

Applicable to extensions involving roots of unity.

Abstract

We deduce a formula enumerating the isomorphism classes of extensions of a $\kp$-adic field $K$ with given ramification $e$ and inertia $f$. The formula follows from a simple group-theoretic lemma, plus the Krasner formula and an elementary class field theory computation. It shows that the number of classes only depends on the ramification and inertia of the extensions $K/\Q_p$, and $K(\zeta_{p^m})/K$ obtained adding the $p^m$-th roots of 1, for all $p^m$ dividing $e$.