Galois module structure of the units modulo $p^m$ of cyclic extensions of degree $p^n$

TL;DR

This paper determines the module structure of units modulo p^m in cyclic extensions of degree p^n, revealing that most indecomposables are cyclic and free, with finitely many classes for fixed parameters, aiding Galois theory insights.

Contribution

It provides a complete description of the Galois module structure of units modulo p^m in cyclic p^n-extensions, including classification of indecomposables and their properties.

Findings

Most indecomposable summands are cyclic and free over quotient groups.

Finitely many isomorphism classes for non-free indecomposables when m,n are fixed.

The results facilitate understanding of Galois cohomology and inverse Galois problems.

Abstract

Let $p$ be prime, and $n,m \in \mathbb{N}$. When $K/F$ is a cyclic extension of degree $p^n$, we determine the $\mathbb{Z}/p^m\mathbb{Z}[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times p^m}$. With at most one exception, each indecomposable summand is cyclic and free over some quotient group of $\text{Gal}(K/F)$. For fixed values of $m$ and $n$, there are only finitely many possible isomorphism classes for the non-free indecomposable summand. These Galois modules act as parameterizing spaces for solutions to certain inverse Galois problems, and therefore this module computation provides insight into the structure of absolute Galois groups. More immediately, however, these results show that Galois cohomology is a context in which seemingly difficult module decompositions can practically be achieved: when $m,n>1$ the modular representation theory allows for an infinite number of indecomposable summands (with no known classification of indecomposable types), and yet the main result of this paper provides a complete decomposition over an infinite family of modules.