  By KMS-classification theorem, the Dedekind zeta function is an invariant of
Bost-Connes systems. In this paper, we show that it is in fact an invariant of
Bost-Connes $C^*$-algebras. We examine second maximal primitive ideals of
Bost-Connes $C^*$-algebras, and apply $K$-theory to some quotients.
