We prove that the Krull-Schmidt decomposition of the Galois module of the $p$-adic completion of algebraic units is controlled by the primes that are ramified in the Galois extension and the $S$-ideal class group. We also compute explicit upper bounds for the number of possible Galois module structures of algebraic units when the Galois group is cyclic of order $p^{ 2}$ or $p^{ 3}$.
Categories: math.NT