Adele ring

Summary

The adele ring $\mathbb{A}_K$ of a number field $K$ is a restricted product of the completions $K_v$ of $K$ at all its places. The restriction is taken with respect to the rings of integers of the non-archimedean completions. It is a locally compact topological ring in which $K$ sits diagonally as a discrete subring with compact quotient. It allows all completions of $K$ to be handled at once. It is the setting of Tate's thesis, of the idele class group form of class field theory and of the theory of automorphic forms.