---
tag: SEH68BA1QQ
kind: concept
name: Adele ring
msc: [11R56]
sources:
  - url: https://en.wikipedia.org/wiki/Adele_ring
same_as:
  wikidata: Q355356
  wikipedia: 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|BT5V3TSNY6]].
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|9GHD9MXNYV]] form of [[class field theory|BEKCEYHCHX]] and of the theory of automorphic forms.

## Explanations

### L1 human

A rational number can be seen inside the real numbers and inside each field of [[p-adic numbers|BT6T702C28]] $\Q_p$.
An adele of $\Q$ is a family with one real number and one $p$-adic number for each prime $p$.
All but finitely many of the $p$-adic entries must be $p$-adic integers.
For example, $1/6$ gives the adele whose entries are all $1/6$; it is allowed, since $1/6$ is a $p$-adic integer for every $p$ except $2$ and $3$.
Working with all these entries together turns many local statements into one global statement.