Class field theory

Summary

Class field theory describes the abelian extensions of a number field $K$ in terms of arithmetic data of $K$ itself. The finite abelian extensions of $K$ correspond to subgroups of generalised ideal class groups. Equivalently, they correspond to open subgroups of finite index of the idele class group. The Artin reciprocity law gives an isomorphism between the Galois group of such an extension and the corresponding quotient, through Frobenius elements. The Kronecker–Weber theorem and the Hilbert class field are special cases, and local class field theory treats the completions.