In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory that describes abelian Galois extensions of local and global fields using objects associated to the ground field. where denotes the idelic norm map from L to F. This isomorphism is named the reciprocity map. The existence theorem states that the reciprocity map can be used to give a bijection between the set of abelian extensions of F and the set of closed subgroups of finite index of Inside class field theory one can distinguish special class field theory and general class field theory.