a basis of over For each finite place of let be the -module generated by in There exists a finite set of places containing such that for all is a compact subring of Furthermore, contains For each is an open subset of and the map is continuous on As a consequence maps homeomorphically on its image in For each the are the elements of mapping in with the function above. Therefore, is an open and compact subgroup of Alternative characterisation of the idele group Proposition. Let be a finite set of places. Then is an open subgroup of where is the union of all Corollary. In the special case of for each finite set of places is an open subgroup of Furthermore, is the union of all Norm on the idele group We want to transfer the trace and the norm from the adele ring to the idele group. It turns out the trace can't be transferred so easily. However, it is possible to transfer the norm from the adele ring to the idele group. Let Then and therefore, we have in injective group homomorphism Since it is invertible, is invertible too, because Therefore As a consequence, the restriction of the norm-function introduces a continuous function: The Idele class group Lemma. There is natural embedding of into given by the diagonal map: Proof. Since is a subset of for all the embedding is well-defined and injective. Corollary. is a discrete subgroup of Defenition. In analogy to the ideal class group, the elements of in are called principal ideles of The quotient group is called idele class group of This group is related to the ideal class group and is a central object in class field theory. Remark. is closed in therefore is a locally compact topological group and a Hausdorff space. Lemma. Let be a finite extension. The embedding induces an injective map: Properties of the idele group Absolute value on the idele group of K and 1-idele Definition. For define: Since is an idele this product is finite and therefore well-defined. Remark. The definition can be extended to by allowing infinite products. However, these infinite products vanish and so vanishes on We will use to denote both the function on and Theorem. is a continuous group homomorphism. Proof. Let where we use that all products are finite. The map is continuous which can be seen using an argument dealing