not a restricted product. Remark. Global function fields do not have any infinite places and therefore the finite adele ring equals the adele ring. Lemma. There is a natural embedding of into given by the diagonal map: Proof. If then for almost all This shows the map is well-defined. It is also injective because the embedding of in is injective for all Remark. By identifying with its image under the diagonal map it is regarded as a subring of The elements of are called the principal adeles of Definition. Let be a set of places of Define the set of the -adeles of as Furthermore, if the result is: The adele ring of rationals By Ostrowski's theorem the places of are it is possible to identify a prime with the equivalence class of the -adic absolute value and with the equivalence class of the absolute value defined as: The completion of with respect to the place is with valuation ring For the place the completion is Thus: Or for short the difference between restricted and unrestricted product topology can be illustrated using a sequence in : Lemma. Consider the following sequence in : In the product topology this converges to , but it does not converge at all in the restricted product topology. Proof. In product topology convergence corresponds to the convergence in each coordinate, which is trivial because the sequences become stationary. The sequence doesn't converge in restricted product topology, for each adele and for each restricted open rectangle we have: for and therefore for all As a result for almost all In this consideration, and are finite subsets of the set of all places. Alternative definition for number fields Definition (profinite integers). The profinite integers are defined as the profinite completion of the rings with the partial order i.e., Lemma. Proof. This follows from the Chinese Remainder Theorem. Lemma. Proof. Use the universal property of the tensor product. Define a -bilinear function This is well-defined because for a given with co-prime there are only finitely many primes dividing Let be another -module with a -bilinear map It must be the case that factors through uniquely, i.e., there exists a unique -linear map such that We define as follows: for a given there exist and such that for all Define One can show is well-defined, -linear, satisfies and is unique with these properties. Corollary. Define This results in an