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We complete the classification of Bost--Connes systems. We show that twoBost--Connes C*-algebras for number fields are isomorphic if and only if theoriginal semigroups actions are conjugate. Together with recent reconstructionresults in number theory by Cornelissen--de Smit--Li--Marcolli--Smit, weconclude that two Bost--Connes C*-algebras are isomorphic if and only if theoriginal number fields are isomorphic.

Reconstructing the Bost--Connes semigroup actions from K-theory