Galois module structure of square power classes for biquadratic extensions

Abstract

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. The example and methodology we consider pave the way for a paradigm shift in the study of absolute Galois groups via Galois modules. Whereas generic modules of this type are completely intractable to approach, this work shows that Galois cohomology provides a context where this otherwise impossible problem becomes approachable, interesting, and illuminating.Let V = Z/2Z⊕Z/2Z. For a Galois extension K/F with char(K) = 2 and Gal(K/F ) V , we determine the F 2 [Gal(K/F )]-module structure of K × /K ×2 . Although there are an infinite number of (pairwise non-isomorphic) indecomposable F 2 [V ]-modules, our decomposition includes at most 8 indecomposable types. This is a dramatic development on several fronts. Most conspicuously, it shows that these modules can be computed in settings which might initially seem too forbidding. The elegant nature of this decomposition also points to new structural restrictions on absolute Galois groups. More generally, the techniques used in this paper provide a blueprint for studying a broad class of Galois modules. This opens the door for a novel, systematic study of the Galois theory of maximal p-extensions of fields.Prepare for the unknown by studying how others in the past have coped with the unforeseeable and the unpredictable.