Quaternion algebras and square power classes over biquadratic extensions

TL;DR

This paper investigates the structure of square power classes over biquadratic extensions, revealing a limited set of summand types and linking their multiplicities to Brauer group subspaces.

Contribution

It determines the multiplicities of summand types in square power classes over biquadratic extensions and relates them to specific subspaces of the Brauer group.

Findings

At most 9 summand types in the decomposition

Multiplicity of summands linked to Brauer group subspaces

All 'unexceptional' summand types are realizable

Abstract

Recently the Galois module structure of square power classes of a field $K$ has been computed under the action of $\text{Gal}(K/F)$ in the case where $\text{Gal}(K/F)$ is the Klein $4$-group. Despite the fact that the modular representation theory over this group ring includes an infinite number of non-isomorphic indecomposable types, the decomposition for square power classes includes at most $9$ distinct summand types. In this paper we determine the multiplicity of each summand type in terms of a particular subspace of $\text{Br}(F)$, and show that all "unexceptional" summand types are possible.