On wild extensions of a p-adic field

TL;DR

This paper classifies degree p^k extensions of p-adic fields without intermediate fields, linking them to Kummer extensions and Galois group representations, and counts their isomorphism classes for specific cases.

Contribution

It establishes a novel correspondence between these extensions and Kummer extensions, and provides explicit counts of isomorphism classes based on Galois groups.

Findings

Classifies extensions via Kummer theory and Galois representations.

Provides counts of isomorphism classes for degree p^2 extensions.

Determines the total number of such extensions.

Abstract

In this paper we consider the problem of classifying the isomorphism classes of extensions of degree pk of a p-adic field, restricting to the case of extensions without intermediate fields. We establish a correspondence between the isomorphism classes of these extensions and some Kummer extensions of a suitable field F containing K. We then describe such classes in terms of the representations of Gal(F/K). Finally, for k = 2 and for each possible Galois group G, we count the number of isomorphism classes of the extensions whose normal closure has a Galois group isomorphic to G. As a byproduct, we get the total number of isomorphism classes.