adèle

Let $K$ be a number field. For each finite prime $v$ of $K$ , let $\mathfrak{o}_{v}$ denote the valuation ring of the completion $K_{v}$ of $K$ at $v$ . The adèle group $\mathbb{A}_{K}$ of $K$ is defined to be the restricted direct product of the collection of locally compact additive groups $\{K_{v}\}$ over all primes $v$ of $K$ (both finite primes and infinite primes), with respect to the collection of compact open subgroups $\{\mathfrak{o}_{v}\}$ defined for all finite primes $v$ .

The set $\mathbb{A}_{K}$ inherits addition and multiplication operations (defined pointwise) which make it into a topological ring. The original field $K$ embeds as a ring into $\mathbb{A}_{K}$ via the map

x\mapsto\prod_{v}x_{v}.

defined for $x\in K$ , where $x_{v}$ denotes the image of $x$ in $K_{v}$ under the embedding $K\hookrightarrow K_{v}$ . Note that $x_{v}\in\mathfrak{o}_{v}$ for all but finitely many $v$ , so that the element $x$ is sent under the above definition into the restricted direct product as claimed.

It turns out that the image of $K$ in $\mathbb{A}_{K}$ is a discrete set and the quotient group $\mathbb{A}_{K}/K$ is a compact space in the quotient topology.

Title	adèle
Canonical name	Adele
Date of creation	2013-03-22 12:39:31
Last modified on	2013-03-22 12:39:31
Owner	djao (24)
Last modified by	djao (24)
Numerical id	5
Author	djao (24)
Entry type	Definition
Classification	msc 11R56
Related topic	Idele
Defines	adèle group
Defines	group of adèles