Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 123 / 4 / 1

The adele ring is the restricted productAddition and multiplication are componentwise. If $x_v,y_v\in {\mathcal{O}}_v$ outside finite sets, then $x_v+y_v,x_v{y}_v\in {\mathcal{O}}_v$ outside their union, so these operations are closed on ${\mathbb{A}}_K$. The componentwise ring axioms make it a ring. Its restricted product topology has basic open sets ${\prod }_v{U}_v$ with each $U_v\subseteq K_v$ open and $U_v={\mathcal{O}}_v$ at all but finitely many finite places.

The idele group isthe restricted product with respect to the local unit groups.

Underthe preimage of a basic product neighbourhood $U\times V$ isAt almost all finite places, $U_v=V_v={\mathcal{O}}_v$, and the condition $x_v,x_v^{-1}\in {\mathcal{O}}_v$ is exactly $x_v\in {\mathcal{O}}_v^{\times }$. Thus these preimages form the usual restricted-product basis for ${\mathbb{I}}_K$. Conversely, every basic idele neighbourhood is obtained by choosing suitable $U_v$ and $V_v$ locally. Hence the restricted product topology on the idele group equals the subspace topology induced by $\varphi$.