Class field theory is a branch of algebraic number theory that explores the connections between number fields and their algebraic structure through the lens of Galois theory. It primarily aims to study abelian extensions of number fields, which are extensions of number fields that are Galois with an abelian Galois group. The theory provides a correspondence between the ideals of a number field and the abelian extensions of that field.
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Global class field theory classifies finite abelian extensions of a number field using ideal-class or idele-class quotients and the Artin reciprocity map.