Class Field Theory

A body of results that classifies abelian extensions of global and local fields in terms of arithmetic data of the base field, giving explicit reciprocity laws and a correspondence between finite abelian extensions and open subgroups of a suitable arithmetic group (idele class group or ray class groups).

The body of results that classifies abelian (commutative Galois) extensions of global and local fields in terms of arithmetic data of the base field, most conveniently expressed via ideles, the idele class group, and reciprocity maps.