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

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

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

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

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

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

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

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

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.