Multiplicative Galois Module Structure (reviewed)

Let $K$ be a number field with Galois closure $K^{gal}$ and $U$ be the group of units of the ring of integers of $K^{gal}$. Let $\mu \subset U$ be the subgroup of roots of unity which lie in $K^{gal}$. Let $M=U/\mu$ be the units modulo its torsion subgroup. This is a lattice which is a module for $\Z[G]$ where $G=\textrm{Gal}(K^{gal}/\Q)$. We give a decomposition of $M$ as a direct sum of indecomposable submodules.