Galois module structure of local unit groups

TL;DR

This paper investigates the structure of local unit groups in finite abelian extensions of p-adic fields, providing explicit module generators over the Galois group ring, with a focus on eigenspaces related to the Teichmüller character.

Contribution

It explicitly determines generators of unit groups as modules over the Galois group ring in abelian p-adic extensions, advancing understanding of their Galois module structure.

Findings

Explicit generators of unit groups as modules over the Galois group ring

Analysis of eigenspaces for powers of the Teichmüller character

Application to the field of norms for p-power roots of unity

Abstract

We study the groups in the unit filtration of a finite abelian extension K of the field of p-adic numbers. We determine explicit generators of these groups as modules over the pro-p group ring of the Galois group of K over the p-adic numbers. We work in eigenspaces for powers of the Teichmueller character, first at the level of the field of norms for the extension of K by p-power roots of unity and then at the level of K.