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 \(p^{3}\).