Quaternion algebras and square power classes over biquadratic extensions

Recently the Galois module structure of square power classes of a field K has been computed under the action of Gal(K/F ) in the case where Gal(K/F ) is the Klein 4group. 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 Br(F ), and show that all “unexceptional" summand types are possible.

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