is a global field, meaning it is either a number field (a finite extension of ) or a global function field (a finite extension of for prime and ). By definition a finite extension of a global field is itself a global field. Valuations For a valuation of it can be written for the completion of with respect to If is discrete it can be written for the valuation ring of and for the maximal ideal of If this is a principal ideal denoting the uniformizing element by A non-Archimedean valuation is written as or and an Archimedean valuation as Then assume all valuations to be non-trivial. There is a one-to-one identification of valuations and absolute values. Fix a constant the valuation is assigned the absolute value defined as: Conversely, the absolute value is assigned the valuation defined as: A place of is a representative of an equivalence class of valuations (or absolute values) of Places corresponding to non-Archimedean valuations are called finite, whereas places corresponding to Archimedean valuations are called infinite. Infinite places of a global field form a finite set, which is denoted by Define and let be its group of units. Then Finite extensions Let be a finite extension of the global field Let be a place of and a place of If the absolute value restricted to is in the equivalence class of , then lies above which is denoted by and defined as: (Note that both products are finite.) If , can be embedded in Therefore, is embedded diagonally in With this embedding is a commutative algebra over with degree The adele ring The set of finite adeles of a global field denoted is defined as the restricted product of with respect to the It is equipped with the restricted product topology, the topology generated by restricted open rectangles, which have the following form: where is a finite set of (finite) places and are open. With component-wise addition and multiplication is also a ring. The adele ring of a global field is defined as the product of with the product of the completions of at its infinite places. The number of infinite places is finite and the completions are either or 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 In the following, we write although this is generally