Abstract: In this paper we use the Merkurjev-Suslin theorem to explore the structure of arithmetically significant Galois modules that arise from Kummer theory. Let $K$ be a field of characteristic different from a prime $\ell$, $n$ a positive integer, and suppose that $K$ contains the $(\ell^n)$th roots of unity. Let $L$ be the maximal $\mathbb{Z}/\ell^n$-elementary abelian extension of $K$, and set $G = \operatorname{Gal}(L|K)$. We consider the $G$-module $J = L^\times/\ell^n$ and denote its socle series by $J_m$. We provide a precise condition, in terms of a map to $H^3(G,\mathbb{Z}/\ell^n)$, determining which submodules of $J_{m-1}$ embed in cyclic modules generated by elements of $J_m$. This generalizes a theorem of Adem, Gao, Karaguezian, and Minac which deals with the case $m=\ell^n=2$. This description of $J_m/J_{m-1}$ can be viewed as an analogue of the classical Hilbert's Theorem 90 and it is helpful for understanding the $G$-module $J$.