On Krull–Schmidt decompositions of unit groups of number fields
Acta Arithmetica
MSC: Primary 11R27; Secondary 11R33, 20C10
DOI: 10.4064/aa240314-24-8
Published online: 25 November 2024
Abstract
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 $p^{3}$.
