Extensions of degree p^l of a p-adic field

TL;DR

This paper classifies and counts specific degree p^l extensions of a p-adic field with no intermediate fields, analyzing their Galois groups, ramification, and discriminants using a novel correspondence with irreducible modules.

Contribution

It introduces a new correspondence between degree p^l extensions with no intermediate fields and irreducible modules, enabling detailed counting and analysis of these extensions.

Findings

Count of isomorphism classes of p^l-extensions without intermediate fields.

Classification of Galois groups of the normal closures.

Determination of ramification groups and discriminants.

Abstract

Given a $p$-adic field $K$ and a prime number $\ell$, we count the total number of the isomorphism classes of $p^\ell$-extensions of $K$ having no intermediate fields. Moreover for each group that can appear as Galois group of the normal closure of such an extension, we count the number of isomorphism classes that contain extensions whose normal closure has Galois group isomorphic to the given group. Finally we determine the ramification groups and the discriminant of the composite of all $p^\ell$-extensions of K with no intermediate fields. The principal tool is a result, proved at the beginning of the paper, which states that there is a one-to-one correspondence between the isomorphism classes of extensions of degree $p^\ell$ of $K$ having no intermediate extensions and the irreducible $H$-submodules of dimension $\ell$ of $F^*/{F^*}^p$, where $F$ is the composite of certain fixed normal extensions of $K$ and $H$ is its Galois group over $K$.