Idele Group

In algebraic number theory, the idele group of an algebraic number field K is the group of invertible elements of the adele ring A_K_. It consists of tuples (a_v_) where each a_v_ is a unit in the completion K_v_, and a_v_ is a unit in the local ring of integers for all but finitely many places v. The idele group, denoted I_K_, is the natural multiplicative counterpart to the additive adele ring, and it carries a topology that makes it a locally compact topological group. The quotient of the idele group by the diagonal embedding of K^× — the idele class group C_K_ — is the central object of global class field theory, where it is identified via the Artin reciprocity map with the abelianized absolute Galois group of K.

The idele group is not merely the multiplicative group of the adele ring. It is the geometric object that reveals why the multiplicative structure of a number field is deeper than its additive structure. The adele ring is a vector space; the idele group is a symmetry group. And in mathematics, symmetries always know more than vectors.