The multiplicative group of an algebraic number field acts by multiplication on the adele ring of the field, and the quotient space for this action is Connes’ adele class space. I will give an overview of joint work with Takuya Takeishi in which we prove that the crossed product C*-algebra associated with the adele class space completely remembers the number field. Precisely, we prove that two such crossed product C*-algebras are *-isomorphic if and only if the underlying number fields are isomorphic. Primitive ideals and subquotients play a central role in our proof.