##  [Artin Reciprocity](/artin-reciprocity-1) 

 Definition

A foundational statement of global class field theory giving a canonical isomorphism (or surjective reciprocity map) between the idele class group of a global field and the abelianized absolute Galois group, identifying arithmetic data with characters and Frobenius elements.

 

 

 

 

 

 





## Principle

Principle

Abelian extensions of a global field correspond to open subgroups of the idele class group: local Frobenius elements at primes match idelic classes and characters on the idele class group parameterize abelian Galois characters, realizing a reciprocity law.

 

 

 

 

 





## Demonstration

Demonstration

Over the rational numbers, Artin reciprocity identifies the quotient of the ideles by Q* with the Galois group of the maximal abelian extension; concretely, splitting of primes in abelian extensions is governed by the image of local uniformizers under the reciprocity map (Frobenius elements).

 

 

 

 

## Misapplication

Misapplication

Applying Artin reciprocity to non-abelian extensions or expecting a direct analogue for arbitrary Galois groups: the theorem is strictly about abelianized Galois groups and must not be used to infer non-abelian correspondences without further structure.

 

 

 

 

 





## Consequence

Consequence

Provides a complete description of abelian extensions of global fields, underpins the formulation of conductors and L-functions for characters, and supplies the global reciprocity constraint used to compute splitting behavior of primes.

 

 

 

 

## Reversal

Reversal

Local reciprocity (the inverse local maps between local multiplicative groups and local Galois groups) complements global reciprocity, but a full non-abelian reversal—replacing idele classes by non-abelian objects to recover full Galois groups—remains outside classical class field theory and is the subject of deeper conjectures.

 

 

 

 

 





## Boundary

Boundary

Applies to global fields (number fields and function fields of curves over finite fields) and to their abelian extensions; it excludes non-abelian extensions and requires working with ideles, completions, and topological notions of openness for the correspondence.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Sits adjacent to the Langlands program and reciprocity conjectures: the tension is between the concrete, abelian classification by ideles and the broader, conjectural non-abelian reciprocity that seeks analogous parametrizations by automorphic data.

 

 

 

 

 





## Synthesis

Synthesis

Artin reciprocity is the classical bijective dictionary of abelian class field theory: it identifies the arithmetic of abelian extensions with characters of the idele class group, encoding splitting and Frobenius behavior and forming the abelian core against which non-abelian generalizations are measured.