A body of results that classifies abelian extensions of global and local fields in terms of arithmetic data of the base field, giving explicit reciprocity laws and a correspondence between finite abelian extensions and open subgroups of a suitable arithmetic group (idele class group or ray class groups).

Principle

The organizing principle is Artin reciprocity: a canonical surjective reciprocity map from an arithmetic group (idele class group in global fields or multiplicative group modulo norms in local fields) onto the abelianized Galois group identifies subgroups with abelian extensions and encodes local-global compatibility and conductors.

Demonstration

The Hilbert class field of a number field is the maximal unramified abelian extension whose Galois group is isomorphic to the class group of the base field; computing this extension realizes ideal-class theoretic arithmetic as a concrete field extension, an illustrative global example.

Misapplication

Treating Class Field Theory as describing non-abelian extensions or expecting explicit generators for arbitrary fields without the abelian hypothesis; many phenomena in non-abelian Galois theory fall outside its scope and require the Langlands program or other non-abelian methods.

Consequence

Provides explicit descriptions of abelian Galois groups in terms of arithmetic invariants, yields reciprocity laws (generalizing quadratic reciprocity), and supplies tools to construct and analyze abelian extensions, conductors, and L-functions central to algebraic number theory.

Reversal

Given knowledge of the abelian Galois group or a specific abelian extension, one can deduce arithmetic properties of the base field (class group, ramification behavior, conductors) by reading off how subgroups correspond to intermediate fields via the reciprocity map.

Boundary

Applies only to abelian (commutative) extensions of local or global fields; it does not classify non-abelian extensions and typically requires the language of ideles, ray class groups, or local norm groups to state precise correspondences and conductors.

Semantic Tension

Stands in contrast with the broader Langlands program, which aims to relate non-abelian extensions and automorphic representations; Class Field Theory is the abelian cornerstone and is sometimes conflated with general reciprocity ambitions that require deeper, non-abelian machinery.

Synthesis

Class Field Theory is the arithmetic reciprocity framework that identifies abelian Galois extensions of a field with explicit subgroups in its idele or ray-class structures via Artin reciprocity, converting ideal-theoretic and local data into a classification of abelian extensions.