Idele group

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 $K$ is the restricted direct product $\mathbb {A} _{K}^{\times }=\prod _{v}'K_{v}^{\times }$ of the multiplicative groups of the completions of $K$, taken with respect to the unit groups ${\mathcal {O}}_{v}^{\times }$ at the non-archimedean places. Equivalently, it is the group of invertible elements of the adele ring $\mathbb {A} _{K}$, equipped with a topology finer than the subspace topology inherited from $\mathbb {A} _{K}$.

The quotient $C_{K}=\mathbb {A} _{K}^{\times }/K^{\times }$ 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 $L$-functions and Hecke characters.