On Krull–Schmidt decompositions of unit groups of number fields

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.