The set of infinite places of a global field is finite, we denote this set by formula_31  Define formula_32 and let formula_33 be its group of units. Then formula_34  Let formula_35 be a finite extension of the global field formula_30 Let formula_37 be a place of formula_38 and formula_8 a place of formula_30 We say formula_37 lies above formula_42 denoted by formula_43 if the absolute value formula_44 restricted to formula_1 is in the equivalence class of formula_12 Define   Note that both products are finite. If formula_48 we can embed formula_10 in formula_50 Therefore, we can embed formula_10 diagonally in formula_52 With this embedding formula_53 is a commutative algebra over formula_10 with degree   The set of finite adeles of a global field formula_56 denoted formula_57 is defined as the restricted product of formula_10 with respect to the formula_59  It is equipped with the restricted product topology, the topology generated by restricted open rectangles, which have the following form:  where formula_62 is a finite set of places and formula_63 are open. With component-wise addition and multiplication formula_64 is also a ring. The adele ring of a global field formula_1 is defined as the product of formula_64 with the product of the completions of formula_1 at its infinite places. The number of infinite places is finite and the completions are either formula_68 or formula_69 In short:  With addition and multiplication defined as component-wise the adele ring is a ring. The elements of the adele ring are called adeles of formula_30 In the following, we write  although this is generally not a restricted product. Remark. Global function fields do not have any infinite places and therefore the finite adele ring equals the adele ring. Proof. If formula_76 then formula_77 for almost all formula_12 This shows the map is well-defined. It is also injective because the embedding of formula_1 in formula_10 is injective for all formula_12   Remark. By identifying formula_1 with its image under the diagonal map we regard it as a subring of formula_83 The elements of formula_1 are called the principal adeles of formula_83  Definition. Let formula_86 be a set of places of formula_30 Define the set of the formula_86-adeles of formula_1 as  Furthermore if we define   we have: formula_92  By Ostrowski's theorem the places of formula_4 are formula_94 where we identify a prime formula_6 with the equivalence class of the formula_6-adic absolute value and formula_97 with the equivalence class of the absolute value formula_98 defined as:  The completion of formula_4 with respect to the place formula_6 is formula_102 with valuation ring formula_103 For the place formula_97 the completion is formula_105 Thus:   Or for short  We will illustrate the difference between restricted and unrestricted product topology using a sequence in formula_108:   Proof.