  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.
