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In mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles) is a central object of class field theory, a branch of algebraic number theory. It is the restricted product of all the completions of the global field and is an example of a self-dual topological ring. An adele derives from a particular kind of idele. "Idele" derives from the French "idèle" and was coined by the French mathematician Claude Chevalley. The word stands for 'ideal element' (abbreviated: id.el.). Adele (French: "adèle") stands for 'additive idele' (that is, additive ideal element). The ring of adeles allows one to describe the Artin reciprocity law, which is a generalisation of quadratic reciprocity, and other reciprocity laws over finite fields. In addition, it is a classical theorem from Weil that -bundles on an algebraic curve over a finite field can be described in terms of adeles for a reductive group . Adeles are also connected with the adelic algebraic groups and adelic curves. The study of geometry of numbers over the ring of adeles of a number field is called adelic geometry. Let be a global field (a finite extension of or the function field of a curve over a finite field). The adele ring of is the subring consisting of the tuples where lies in the subring for all but finitely many places . Here the index ranges over all valuations of the global field , is the completion at that valuation and the corresponding valuation ring. The ring of adeles solves the technical problem of "doing analysis on the rational numbers ." The classical solution was to pass to the standard metric completion and use analytic techniques there. But, as was learned later on, there are many more absolute values other than the Euclidean distance, one for each prime number , as was classified by Ostrowski. The Euclidean absolute value, denoted , is only one among many others, , but the ring of adeles makes it possible to compromise and .

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