How is class field theory related to quadratic reciprocity?

Quadratic reciprocity is the simplest case: it describes the abelian extension obtained by adjoining a square root, and Artin reciprocity generalizes it to all abelian extensions of any number field.

What is the Hilbert class field?

It is the largest abelian extension of a number field that is unramified everywhere; its Galois group is naturally isomorphic to the field's ideal class group, so its degree equals the class number.

Class Field Theory

Class field theory is the crowning achievement of algebraic number theory: it classifies all abelian extensions of a number field in terms of the field's own arithmetic, generalizing quadratic reciprocity to a sweeping reciprocity law.

Definition

Class field theory establishes a correspondence between the finite abelian extensions of a number field and certain quotient groups of its idele class group (or generalized ideal class groups), with the Artin reciprocity map providing a canonical isomorphism onto each extension's Galois group.

Scope

This topic covers the main theorems of class field theory in their classical and idelic formulations: the Artin reciprocity law and the Artin map from generalized ideal class groups to Galois groups, the existence theorem matching congruence subgroups to abelian extensions, conductors, the Hilbert class field as the maximal unramified abelian extension, the Kronecker-Weber theorem realizing abelian extensions of the rationals inside cyclotomic fields, and the role of local class field theory.

Core questions

How does the Artin map send arithmetic data to Galois automorphisms, and why is it a reciprocity law?
Which subgroups of the idele class group correspond to which abelian extensions (the existence theorem)?
What is the Hilbert class field, and how does its Galois group recover the ideal class group?
How does the Kronecker-Weber theorem describe every abelian extension of the rationals?

Key theories

Artin reciprocity: For an abelian extension, the Artin map sending each unramified prime to its Frobenius extends to an isomorphism from a generalized ideal class group onto the Galois group, a vast generalization of quadratic reciprocity.
Existence theorem and Hilbert class field: Every open subgroup of finite index in the idele class group is the norm group of a unique abelian extension; the Hilbert class field is the maximal unramified one, with Galois group canonically the ideal class group.
Kronecker-Weber theorem: Every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity, the first and prototypical instance of explicit class field theory.

Clinical relevance

Class field theory frames the Langlands program and the modularity results behind the proof of Fermat's Last Theorem; explicit forms, including complex multiplication, also drive constructions used in elliptic-curve and isogeny-based cryptography.

History

Hilbert conjectured the existence of the class field and posed guiding problems around 1900. Takagi proved the existence theorem in 1920, Artin established the reciprocity law in 1927, and Chevalley's introduction of ideles in the 1930s gave the theory its modern adelic form, setting the stage for the Langlands program.

Key figures

David Hilbert
Teiji Takagi
Emil Artin
Helmut Hasse

Seminal works

cox2013

Frequently asked questions

How is class field theory related to quadratic reciprocity?: Quadratic reciprocity is the simplest case: it describes the abelian extension obtained by adjoining a square root, and Artin reciprocity generalizes it to all abelian extensions of any number field.
What is the Hilbert class field?: It is the largest abelian extension of a number field that is unramified everywhere; its Galois group is naturally isomorphic to the field's ideal class group, so its degree equals the class number.