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In mathematics, class field theory 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.
Hilbert is credited as one of pioneers of class field. However, this notion was already familiar for Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin.
One of the major results is: given a number field F, and writing K for the maximal abelian unramified extension of F, the Galois group of K over F is canonically isomorphic to the ideal class group of F. This statement was generalized to the so called Artin reciprocity law; in the idelic language, writing CF for the idele class group of F, and taking L to be any finite abelian extension of F, this law gives a canonical isomorphism
where N L / F {\displaystyle N_{L/F}} denotes the idelic norm map from L to F. This isomorphism is named the reciprocity map.