Galois module structure of some elementary $p$-abelian extensions

TL;DR

This paper investigates the Galois module structure of elementary p-abelian extensions over a field, under specific conditions on the Galois group and roots of unity, extending understanding of such algebraic structures.

Contribution

It determines the Galois module structure of parameterizing spaces for elementary p-abelian extensions with general finite p-group Galois groups, under certain pro-p group assumptions.

Findings

Explicit description of Galois module structure for given conditions

Extension of known results to broader classes of Galois groups

Provides tools for analyzing elementary p-abelian extensions in number theory

Abstract

We determine the Galois module structure of the parameterizing space of elementary $p$-abelian extensions of a field $K$ when $\text{Gal}(K/F)$ is any finite $p$-group, under the assumption that the maximal pro-$p$ quotient of the absolute Galois group of $F$ is a free, finitely generated pro-$p$ group, and that $F$ contains a primitive $p$th root of unity if $\text{char}(F) \neq p$.