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

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.