Acronyms that contain the term class field theory 

Class field theory

In mathematics, class field theory is the branch of algebraic number theory concerned with describing the Galois extensions of local and global fields. Hilbert is often credited for the notion of class field. But it was already familiar for Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. This theory has its origins in the proof of quadratic reciprocity by Gauss at the end of the 18th century. These ideas were developed over the next century, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin. These conjectures and their proofs constitute the main body of class field theory. One major result states that, 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 can be generalized to the Artin reciprocity law; 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 θ L / F : C F / N L / F ( C L ) → Gal ⁡ ( L / F ) , {\displaystyle \theta _{L/F}:C_{F}/{N_{L/F}(C_{L})}\to \operatorname {Gal} (L/F),} where N L / F {\displaystyle N_{L/F}} denotes the idelic norm map from L to F. This isomorphism is then called 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 C F . {\displaystyle C_{F}.} A standard method for developing global class field theory since the 1930s is to develop local class field theory, which describes abelian extensions of local fields, and then use it to construct global class field theory. This was first done by Artin and Tate using the theory of group cohomology, and in particular by developing the notion of class formations. Later, Neukirch found a proof of the main statements of global class field theory without using cohomological ideas. Class field theory also encompasses the explicit construction of maximal abelian extensions of number fields in the few cases where such constructions are known. Currently, this portion of the theory consists of Kronecker–Weber theorem, which can be used to construct the abelian extensions of Q {\displaystyle \mathbb {Q} } , and the theory of complex multiplication, which can be used to construct the abelian extensions of CM-fields. The Langlands program gives one approach for generalizing class field theory to non-abelian extensions. This generalization is mostly still conjectural. For number fields, class field theory and the results related to the modularity theorem are the only cases known.

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