Let K be a number fieldMathworldPlanetmath. For each finite prime v of K, let 𝔬v denote the valuation ringMathworldPlanetmathPlanetmath of the completion Kv of K at v. The adèle group 𝔸K of K is defined to be the restricted direct productPlanetmathPlanetmath of the collection of locally compact additive groupsMathworldPlanetmath {Kv} over all primes v of K (both finite primes and infinite primes), with respect to the collection of compactPlanetmathPlanetmath open subgroups {𝔬v} defined for all finite primes v.

The set 𝔸K inherits addition and multiplication operations (defined pointwise) which make it into a topological ring. The original field K embeds as a ring into 𝔸K via the map


defined for xK, where xv denotes the image of x in Kv under the embedding KKv. Note that xv𝔬v 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 𝔸K is a discrete set and the quotient groupMathworldPlanetmath 𝔸K/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