Global class field theory classifies finite abelian extensions of a number field using ideal-class or idele-class quotients and the Artin reciprocity map.
A modulus is a formal product , where is a nonzero integral ideal and is a product of distinct real places.
The ray class group modulo is , where the numerator consists of fractional ideals prime to and the denominator consists of with at the finite modulus and positive at every real place in .
For an unramified prime in an abelian extension , the Artin symbol is the unique Galois element acting on residue fields by .
The Artin map sends an ideal prime to the ramified primes to the product of its Frobenius elements. For a suitable modulus, its kernel is the norm or congruence subgroup associated with the abelian extension.
Finite abelian extensions of correspond contravariantly to congruence subgroups with , and the Artin map identifies with the Galois group.
The ray class field modulo is the maximal abelian extension whose Artin map factors through the ray class group; its degree is the ray class number.
The ideles congruent to one modulo have finite components in at primes in the finite modulus and positive components at selected real places. Their image in is the idelic congruence subgroup.
For a finite abelian extension , the norm group is the image of the idele-class norm . Class field theory gives .
The Hilbert class field is the maximal unramified abelian extension of a number field. Its Galois group is canonically the ideal class group.
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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.