In number theory, the idele group is a way of packaging the multiplicative arithmetic of a global field at all of its completions at once, so that it contains the information of unique factorization as well as the data relating to units. Formally, the idele group of a global field K {\displaystyle K} is the restricted direct product
A K × = ∏ v ′ K v × {\displaystyle \mathbb {A} _{K}^{\times }=\prod _{v}'K_{v}^{\times }}
of the multiplicative groups of the completions of K {\displaystyle K} , taken with respect to the unit groups O v × {\displaystyle {\mathcal {O}}_{v}^{\times }} at the non-archimedean places. Equivalently, it is the group of invertible elements of the adele ring A K {\displaystyle \mathbb {A} _{K}} , equipped with a topology finer than the subspace topology inherited from A K {\displaystyle \mathbb {A} _{K}} . The quotient C K = A K × / K × {\displaystyle C_{K}=\mathbb {A} _{K}^{\times }/K^{\times }} is the idele class group. Ideles and idele class groups are used in class field theory. They were exploited by John Tate in his thesis to formulate global zeta and L {\displaystyle L} -functions and Hecke characters.
Definition Let K {\displaystyle K} be a global field, and let v {\displaystyle v} run over the places of K {\displaystyle K} . For each place v {\displaystyle v} , let K v {\displaystyle K_{v}} denote the completion of K {\displaystyle K} at v {\displaystyle v} . If v {\displaystyle v} is non-archimedean, let O v {\displaystyle {\mathcal {O}}_{v}} be the corresponding valuation ring and let O v × {\displaystyle {\mathcal {O}}_{v}^{\times }} be its group of units. The idele group of K {\displaystyle K} , usually denoted A K × {\displaystyle \mathbb {A} _{K}^{\times }} or I K {\displaystyle I_{K}} , is the restricted product
A K × = ∏ v ′ K v × {\displaystyle \mathbb {A} _{K}^{\times }=\prod _{v}'K_{v}^{\times }}
of the groups K v × {\displaystyle K_{v}^{\times }} , taken with respect to the subgroups O v × {\displaystyle {\mathcal {O}}_{v}^{\times }} at the non-archimedean places. Thus an idele is a family
x = ( x v ) v , x v ∈ K v × , {\displaystyle x=(x_{v})_{v},\qquad x_{v}\in K_{v}^{\times },}
such that
x v ∈ O v × {\displaystyle x_{v}\in {\mathcal {O}}_{v}^{\times }}
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