The adelic formulation

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Chapter 6 The adelic formulation

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 use in the abstract class field theory we have just set up.

Remark 6.0.1. Spelling note.

There is a lack of consensus regarding the presence or absence of accents in the words adèle and idèle. The term idèle is thought to be a contraction of “ideal element”; it makes its first appearance, with the accent, in Chevalley’s 1940 paper [6]. 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.