History of The Terminology
Historically the idèles were introduced by Chevalley (1936) under the name "élément idéal", which is "ideales Element" in German, which Chevalley (1940) then abbreviated to "idèle". (In these papers he also gave the ideles a rather non-Hausdorff topology.) This was to formulate class field theory for infinite extensions in terms of topological groups. Weil (1938) defined (but did not name) the ring of adeles in the function field case and pointed out that Chevalley's group of "Idealelemente" was the group of invertible elements of this ring. Tate (1950) defined the ring of adeles as a restricted direct product, though he called its elements "valuation vectors" rather than adeles. Chevalley (1951) defined the ring of adeles in the function field case, under the name "repartitions". The term adèle (short for additive idèles, and also a French girls' name) was in use shortly afterwards (Jaffard 1953) and may have been introduced by André Weil. The general construction of adelic algebraic groups by Ono (1957) followed the algebraic group theory founded by Armand Borel and Harish-Chandra.
Read more about this topic: Adelic Algebraic Group
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