When a field $K$ contains a primitive $p$th root of unity, Kummer theory tells us that the $\mathbb{F}_p$-space $K^{\times p}/K^\times$ is a parameterizing space for elementary $p$-abelian extensions of $K$. In previous work, the authors computed the Galois module structure of this set when the Galois group came from an extension $K/F$ whose Galois group is isomorphic to $\mathbb{Z}/p^n\mathbb{Z}$. In this talk we consider the more refined group $K^{\times p^s}/K^\times$ as a Galois module, and we determine its structure. Although the modular representation theory in this case is unwieldy, it turns out that there is only one summand in the decomposition of $K^{\times p^s}/K^\times$ which is not free (either under the full ring or one of its natural quotients). Furthermore, this "exceptional" summand's structure is connected to the cyclotomic character and a certain family of embedding problems along the tower $K/F$. This work is joint with J\'{a}n Min\'{a}\v{c} and John Swallow.