The \(p\)-adic numbers, and more general local fields, were introduced into number theory as a way to translate local facts about number fields (i.e., facts concerning a single prime ideal) into statements of a topological flavor. To prove the statements of class field theory, we need an analogous global construction. To this end, we construct a topological object that includes all of the completions of a number field, including both the archimedean and nonarchimedean ones. This object will be the ring of adèles, and it will lead us to the right target group for a statement of global reciprocity more parallel to the local theory.
Remark6.0.1.Spelling note.
There is a lack of consensus regarding the presence or absence of accents in the words adèle and idèle, in either English or French. The term idèle appears to be a contraction of ideal element; it makes its first appearance, with the accent, in Chevalley’s 1940 paper [6] written in French. The term adèle appeared in the 1950s, possibly as a contraction of additive idèle; it appears to have been suggested by Weil as a replacement for Tate’s term valuation vector and Chevalley’s term repartition. Based on this history, we have opted for the accented spellings here; however, we drop the accent in inflected forms such as adelic.
