Arithmetic properties encoded in the Galois module structure of $K^\times/K^{\times p^m}$

TL;DR

This paper explores the Galois module structure of power classes in fields, revealing that simple decompositions are common but critical non-free summands encode key arithmetic properties, with new interpretations linked to Galois embedding problems.

Contribution

It provides field-theoretic interpretations for parameters in the module decomposition, connecting algebraic structures to Galois embedding problems and cyclotomic characters.

Findings

Decomposition involves at most one non-free summand.

Non-free summands are crucial for understanding field arithmetic.

Parameters relate to solvability of Galois embedding problems.

Abstract

The power classes of a field are well-known for their ability to parameterize elementary $p$-abelian Galois extensions. These classical objects have recently been reexamined through the lens of their Galois module structure. Module decompositions have been computed in several cases, providing deep new insight into absolute Galois groups. The surprising result in each case is that there are far fewer isomorphism types of indecomposables than one would expect generically, with summands predominately free over associated quotient rings. Though non-free summands are the exception both in their form and prevalence, they play the critical role in controlling arithmetic conditions in the field which allow the rest of the decomposition to be so simple. Suppose $m,n \in \mathbb{N}$ and $p$ is prime. In a recent paper, a surprising and elegant decomposition for $p^m$th power classes has been computed when the underlying Galois group is a cyclic group of order $p^n$. As with previous module decompositions, at most one non-free summand appears. Outside of a particular special case when $p=2$, the structure of this exceptional summand was determined by a vector $\mathbf{a}\in \{-\infty,0,\dots,n\}^m$ and a natural number $d$. In this paper we give field-theoretic interpretations for $\mathbf{a}$ and $d$, showing they are related to the solvability of a family of Galois embedding problems and the cyclotomic character associated to $K/F$.