---
tag: 5KMSW7139A
kind: definition
name: Idele class group
msc: [11R56, 11R37]
defines: 9GHD9MXNYV
uses: [MWBDFS2KJ5, BT5V3TSNY6, SEH68BA1QQ, 0KYX2N69J0]
sources:
  - url: https://en.wikipedia.org/wiki/Adelic_algebraic_group
---

## Definition

Let $K$ be a [[number field|MWBDFS2KJ5]].
For each [[place|BT5V3TSNY6]] $v$ 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|0KYX2N69J0]]
$$C_K = I_K / K^\times,$$
with the quotient topology.

## Remarks

Example: for $K = \Q$, the family with $x_\infty = 2$, $x_3 = 1/3$ and $x_p = 1$ for all other $p$ is an idele, because only finitely many entries are not units.
Non-example: the family with $x_p = p$ for every prime $p$ and $x_\infty = 1$ is not an idele, because $x_p \notin \Z_p^\times$ for infinitely many $p$.
The ideles are the units of the [[adele ring|SEH68BA1QQ]] $\mathbb{A}_K$, but the topology above is finer than the subspace topology from $\mathbb{A}_K$.
The idele norm $\abs{x} = \prod_v \abs{x_v}_v$ equals $1$ on $K^\times$ by the product formula, so it is defined on $C_K$.
Its kernel $C_K^1$ is compact.
The [[ideal class group|WQ46BD3ARY]] of $K$ is a quotient of $C_K$.
The same construction works for every [[global field|634XKY8RZ0]].