algebraic isomorphism Proof. Lemma. For a number field Remark. Using where there are summands, give the right side receives the product topology and transport this topology via the isomorphism onto The adele ring of a finite extension If be a finite extension, then is a global field. Thus is defined, and can be identified with a subgroup of Map to where for Then is in the subgroup if for and for all lying above the same place of Lemma. If is a finite extension, then both algebraically and topologically. With the help of this isomorphism, the inclusion is given by Furthermore, the principal adeles in can be identified with a subgroup of principal adeles in via the map Proof. Let be a basis of over Then for almost all Furthermore, there are the following isomorphisms: For the second use the map: in which is the canonical embedding and The restricted product is taken on both sides with respect to Corollary. As additive groups where the right side has summands. The set of principal adeles in is identified with the set where the left side has summands and is considered as a subset of The adele ring of vector-spaces and algebras Lemma. Suppose is a finite set of places of and define Equip with the product topology and define addition and multiplication component-wise. Then is a locally compact topological ring. Remark. If is another finite set of places of containing then is an open subring of Now, an alternative characterization of the adele ring can be presented. The adele ring is the union of all sets : Equivalently is the set of all so that for almost all The topology of is induced by the requirement that all be open subrings of Thus, is a locally compact topological ring. Fix a place of Let be a finite set of places of containing and Define Then: Furthermore, define where runs through all finite sets containing Then: via the map The entire procedure above holds with a finite subset instead of By construction of there is a natural embedding: Furthermore, there exists a natural projection The adele ring of a vector-space Let be a finite dimensional vector-space over and a basis for over For each place of : The adele ring of is defined as This definition is based on the alternative description of the adele ring as a tensor product equipped with