is generated by for some Choose so that then is the direct product of and the subgroup generated by This subgroup is discrete and isomorphic to For define: The map is an isomorphism of in a closed subgroup of and The isomorphism is given by multiplication: Obviously, is a homomorphism. To show it is injective, let Since for it stands that for Moreover, it exists a so that for Therefore, for Moreover implies where is the number of infinite places of As a consequence and therefore is injective. To show surjectivity, let We define and furthermore, we define for and for Define It stands, that Therefore, is surjective. The other equations follow similarly. Characterisation of the idele group Theorem. Let be a number field. There exists a finite set of places such that: Proof. The class number of a number field is finite so let be the ideals, representing the classes in These ideals are generated by a finite number of prime ideals Let be a finite set of places containing and the finite places corresponding to Consider the isomorphism: induced by At infinite places the statement is immediate, so we prove the statement for finite places. The inclusion ″″ is obvious. Let The corresponding ideal belongs to a class meaning for a principal ideal The idele maps to the ideal under the map That means Since the prime ideals in are in it follows for all that means for all It follows, that therefore Applications Finiteness of the class number of a number field In the previous section we used the fact that the class number of a number field is finite. Here we would like to prove this statement: Theorem (finiteness of the class number of a number field). Let be a number field. Then Proof. The map is surjective and therefore is the continuous image of the compact set Thus, is compact. In addition, it is discrete and so finite. Remark. There is a similar result for the case of a global function field. In this case, the so-called divisor group is defined. It can be shown that the quotient of the set of all divisors of degree by the set of the principal divisors is a finite group. Group of units and Dirichlet's unit theorem Let be a finite set of places. Define Then is a subgroup of containing all elements satisfying for all Since is