Galois module structure of square power classes for biquadratic extensions

For a Galois extension K/F with char(K) not equal 2 and Gal(K/F) similar or equal to Z/2f circle plus Z/2Z, we determine the F-2[Gal(K/F)]-module structure of K-x/K-x2. Although there are an infinite number of (pairwise nonisomorphic) indecomposable F-2[Z/2Z circle plus Z/2Z] -modules, our decomposition includes at most nine 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.