2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/141055We give a complete description of the phase transition of the Bost-Connes type systems for number fields recently introduced by Connes-Marcolli-Ramachandran and Ha-Paugam. We also introduce a notion of K-lattices and discuss an interpretation of these systems in terms of 1-dimensional K-lattices.12 pages; small corrections, references addedOperator AlgebrasNumber Theorytext