  We prove that the class of crossed product C*-algebras associated with the
action of the multiplicative group of a number field on its ring of finite
adeles is rigid in the following explicit sense: Given any *-isomorphism
between two such C*-algebras, we construct an isomorphism between the
underlying number fields. As an application, we prove an analogue of the
Neukirch--Uchida theorem using topological full groups, which gives a new class
of discrete groups associated with number fields whose abstract isomorphism
class completely characterises the number field.
