Galois module structure of $p^{\text{th}}$ power classes of abelian extensions of local fields

TL;DR

This paper investigates the Galois module structure of certain classes in local field extensions, extending previous results and computing invariants for odd primes, revealing a constant Jordan type in specific cases.

Contribution

It extends the understanding of Galois module structures of $p^{th}$ power classes in local fields, including computations of invariants for odd primes.

Findings

$J$ has constant Jordan type $[1]^2$ for $p$-elementary abelian extensions

The results generalize previous work for $p=2$ to odd primes

Computed invariants previously known only for $p=2$ for odd primes

Abstract

In this paper, we describe the Galois module structure of $J=\mathbf{K}^{\times}/\mathbf{K}^{\times p}$, where $\mathbf{K}$ is an extension of a local field $\mathbf{k}$ containing a primitive $p$-th root of unity: for instance, if $\mathbf{K}/\mathbf{k}$ is a $p$-elementary abelian extension, we prove that $J$ is a module of constant Jordan type, with stable Jordan type $[1]^2$, which, in a way, extends the result of J. Min\'a\v{c} and J. Swallow. Also, we take profit from our proof by computing some invariants, which were previously introduced by A. Adem, W. Gao, D. B. Karageuzian and J. Min\'a\v{c} only for $p=2$.