Let K be a number fieldMathworldPlanetmath. For each finite prime v of K, let 𝔬v be the valuation ringMathworldPlanetmathPlanetmath of the completion Kv of K at v, and let Uv be the group of units in 𝔬v. Then each group Uv is a compactPlanetmathPlanetmath open subgroup of the group of units Kv* of Kv. The idèle group 𝕀K of K is defined to be the restricted direct productPlanetmathPlanetmath of the multiplicative groupsMathworldPlanetmath {Kv*} with respect to the compact open subgroups {Uv}, taken over all finite primes and infinite primes v of K.

The units K* in K embed into 𝕀K via the diagonal embedding


where xv is the image of x under the embedding KKv of K into its completion Kv. As in the case of adèles, the group K* is a discrete subgroup of the group of idèles 𝕀K, but unlike the case of adèles, the quotient groupMathworldPlanetmath 𝕀K/K* is not a compact group. It is, however, possible to define a certain subgroupMathworldPlanetmathPlanetmath of the idèles (the subgroup of norm 1 elements) which does have compact quotient under K*.

Warning: The group 𝕀K is a multiplicative subgroup of the ring of adèles 𝔸K, but the topologyMathworldPlanetmath on 𝕀K is different from the subspace topology that 𝕀K would have as a subset of 𝔸K.

Title idèle
Canonical name Idele
Date of creation 2013-03-22 12:39:28
Last modified on 2013-03-22 12:39:28
Owner djao (24)
Last modified by djao (24)
Numerical id 7
Author djao (24)
Entry type Definition
Classification msc 11R56
Related topic Adele
Defines idèle group
Defines group of idèles