Class Field Theory

Class field theory is the branch of algebraic number theory that classifies the abelian extensions of a number field in terms of its ideal class group and its generalizations. Developed by David Hilbert, Teiji Takagi, and Emil Artin, it establishes a correspondence between the abelian Galois group of an extension and a quotient of the field's arithmetic structure. The theory is the prototype of a Langlands correspondence, in which the representation theory of a Galois group is matched to the harmonic analysis of an arithmetic group.

Class field theory is often presented as the summit of classical algebraic number theory. It is not; it is the base camp from which the Langlands program ascends. The abelian case was not the whole story; it was the chapter that taught us how to read.