Construction and classification of some Galois modules

TL;DR

This paper completes the classification of Galois modules associated with cyclic extensions by constructing field extensions with specific arithmetic invariants, building on previous work describing their structure.

Contribution

It introduces a method to construct field extensions with prescribed invariants, advancing the classification of Galois modules in cyclic extensions.

Findings

Constructed field extensions with specific arithmetic invariants.

Completed the classification of Galois modules for cyclic extensions.

Extended previous structural descriptions with explicit constructions.

Abstract

In our previous paper we describe the Galois module structures of $p$th-power class groups $K^\times/{K^{\times p}}$, where $K/F$ is a cyclic extension of degree $p$ over a field $F$ containing a primitive $p$th root of unity. Our description relies upon arithmetic invariants associated with $K/F$. Here we construct field extensions $K/F$ with prescribed arithmetic invariants, thus completing our classification of Galois modules $K^{\times}/K^{\times p}$.