the same topology we defined when giving an alternate definition of adele ring for number fields. Next, is equipped with the restricted product topology. Then and is embedded in naturally via the map An alternative definition of the topology on can be provided. Consider all linear maps: Using the natural embeddings and extend these linear maps to: The topology on is the coarsest topology for which all these extensions are continuous. The topology can be defined in a different way. Fixing a basis for over results in an isomorphism Therefore fixing a basis induces an isomorphism The left-hand side is supplied with the product topology and transport this topology with the isomorphism onto the right-hand side. The topology doesn't depend on the choice of the basis, because another basis defines a second isomorphism. By composing both isomorphisms, a linear homeomorphism which transfers the two topologies into each other is obtained. More formally where the sums have summands. In case of the definition above is consistent with the results about the adele ring of a finite extension The adele ring of an algebra Let be a finite-dimensional algebra over In particular, is a finite-dimensional vector-space over As a consequence, is defined and Since there is multiplication on and a multiplication on can be defined via: As a consequence, is an algebra with a unit over Let be a finite subset of containing a basis for over For any finite place , is defined as the -module generated by in For each finite set of places, define One can show there is a finite set so that is an open subring of if Furthermore is the union of all these subrings and for the definition above is consistent with the definition of the adele ring. Trace and norm on the adele ring Let be a finite extension. Since and from the Lemma above, can be interpreted as a closed subring of For this embedding, write . Explicitly for all places of above and for any Let be a tower of global fields. Then: Furthermore, restricted to the principal adeles is the natural injection Let be a basis of the field extension Then each can be written as where are unique. The map is continuous. Define depending on via the equations: Now, define the trace and norm of as: These are the trace and the determinant of the linear map They are continuous