Adele ring

In number theory, the adele ring is a construction that combines all local versions of a global field into one object. For the rational numbers, these local versions include the real numbers and the fields of p {\displaystyle p} -adic numbers for all prime numbers p {\displaystyle p} . More generally, if K {\displaystyle K} is a global field, its adele ring, often denoted A K {\displaystyle \mathbb {A} _{K}} , is a topological ring built from the completions K v {\displaystyle K_{v}} of K {\displaystyle K} at all its places v {\displaystyle v} . Formally, it is a restricted product of the local fields K v {\displaystyle K_{v}} , with respect to the valuation rings at the non-archimedean places. Its elements are called adeles. The restricted product topology makes A K {\displaystyle \mathbb {A} _{K}} a locally compact topological ring. The field K {\displaystyle K} embeds diagonally in A K {\displaystyle \mathbb {A} _{K}} as a discrete subring, and the quotient A K / K {\displaystyle \mathbb {A} _{K}/K} is compact. As an additive locally compact abelian group, the adele ring is self-dual, making it a natural setting for Fourier analysis on global fields. The group of units of the adele ring, with its natural topology, is the idele group A K × {\displaystyle \mathbb {A} _{K}^{\times }} . The quotient A K × / K × {\displaystyle \mathbb {A} _{K}^{\times }/K^{\times }} , called the idele class group, is a central object in class field theory. Adeles and ideles are also used in Tate's thesis, the theory of automorphic forms, local-global principles, and adelic descriptions of divisors, line bundles, and principal bundles on algebraic curves.