Idele class group

Definition

Let $K$ be a number field. For each place $v$ of $K$, let $K_v$ be the completion of $K$ at $v$. For each finite place $v$, let $\mathcal{O}_v$ be the ring of integers of $K_v$ and $\mathcal{O}_v^\times$ its group of units. The idele group $I_K$ is the set of families $x = (x_v)_v$ such that:

1. $x_v \in K_v^\times$ for every place $v$;
2. $x_v \in \mathcal{O}_v^\times$ for all but finitely many finite places $v$.

It is a group under componentwise multiplication. Its topology has as a basis the sets $\prod_{v \in S} U_v \times \prod_{v \notin S} \mathcal{O}_v^\times$. Here $S$ is a finite set of places containing all infinite places, and each $U_v$ is open in $K_v^\times$. The map $a \mapsto (a, a, \ldots)$ embeds $K^\times$ in $I_K$ as a discrete subgroup. The idele class group of $K$ is the quotient group $$C_K = I_K / K^\times,$$ with the quotient topology.