Galois module structure of square power classes for biquadratic extensions

TL;DR

This paper fully determines the Galois module structure of square power classes in biquadratic extensions, revealing a finite classification despite the infinite variety of indecomposable modules.

Contribution

It provides the first complete description of the Galois module structure of power classes in such extensions, limiting the indecomposable types to at most nine.

Findings

Decomposition includes at most 9 indecomposable types.

First complete determination for this class of extensions.

Advances understanding of modular representation theory in Galois modules.

Abstract

For a Galois extension $K/F$ with $\text{char}(K)\neq 2$ and $\text{Gal}(K/F) \simeq \mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$, we determine the $\mathbb{F}_2[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times 2}$. Although there are an infinite number of (pairwise non-isomorphic) indecomposable $\mathbb{F}_2[\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}]$-modules, our decomposition includes at most $9$ indecomposable types. This paper marks the first time that the Galois module structure of power classes of a field has been fully determined when the modular representation theory allows for an infinite number of indecomposable types.