Galois module structure of the units modulo $$p^m$$ of cyclic extensions of degree $$p^n$$

Let p be prime, and $$n,m \in \mathbb {N}$$ . When K/F is a cyclic extension of degree $$p^n$$ , we determine the $$\mathbb {Z}/p^m\mathbb {Z}[\text {Gal}(K/F)]$$ -module structure $$K^\times /K^{\times p^m}$$ With at most one exception, each indecomposable summand free over some quotient group $$\text {Gal}(K/F)$$ For fixed values m n, there are only finitely many possible isomorphism classes for non-free summand. These Galois modules act as parameterizing spaces solutions to certain inverse problems, therefore this module computation provides insight into absolute groups. More immediately, however, these results show that cohomology context in which seemingly difficult decompositions can practically achieved: when $$m,n>1$$ modular representation theory allows an infinite number summands (with no known classification types), yet main result paper complete decomposition family modules.