  The multiplicative group of a number field acts by multiplication on the full
adele ring of the field. Generalising a theorem of Laca and Raeburn, we
explicitly describe the primitive ideal space of the crossed product C*-algebra
associated with this action. We then distinguish real, complex, and finite
places of the number field using K-theoretic invariants. Combining these
results with a recent rigidity theorem of the authors implies that any
*-isomorphism between two such C*-algebras gives rise to an isomorphism of the
underlying number fields that is constructed from the *-isomorphism.
