Editing Adele ring (section)

{{Short description|Central object of class field theory}}
{{About|the concept in mathematics|the singer|Adele}}
In [[mathematics]], the '''adele ring''' of a [[global field]] (also '''adelic ring''', '''ring of adeles''' or '''ring of adèles'''<ref>{{Cite journal|last=Groechenig|first=Michael|date=August 2017|title=Adelic Descent Theory|journal=Compositio Mathematica|volume=153|issue=8|pages=1706–1746|doi=10.1112/S0010437X17007217|issn=0010-437X|arxiv=1511.06271|s2cid=54016389}}</ref>) is a central object of [[class field theory]], a branch of [[algebraic number theory]]. It is the [[restricted product]] of all the [[Complete metric space|completions]] of the global field and is an example of a [[Duality (mathematics)|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 law]]s over [[List of mathematical jargon|finite]] fields. In addition, it is a classical [[theorem]] from [[André Weil|Weil]] that [[Torsor (algebraic geometry)|<math>G</math>-bundles]] on an [[algebraic curve]] over a finite field can be described in terms of adeles for a [[reductive group]] [[Torsor (algebraic geometry)|<math>G</math>]]. Adeles are also connected with the [[adelic algebraic group]]s and adelic curves.

The study of [[geometry of numbers]] over the ring of adeles of a [[number field]] is called '''adelic geometry'''.