The body of results that classifies abelian (commutative Galois) extensions of global and local fields in terms of arithmetic data of the base field, most conveniently expressed via ideles, the idele class group, and reciprocity maps.

Principle

Establish a natural isomorphism between quotients of the idele class group (or ray class groups) and the Galois groups of maximal abelian extensions, encoding how local information and global norms control abelian extensions through reciprocity laws.

Demonstration

The classical statement: for a number field K, the Artin reciprocity map gives a continuous surjection from the idele class group of K onto Gal(K^ab/K) with kernel equal to the connected component coming from global norms; the Hilbert class field is the maximal unramified abelian extension corresponding to the ideal class group.

Misapplication

Trying to apply abelian reciprocity statements directly to nonabelian extensions or expecting explicit generation of nonabelian extension fields from idele data, which overstates the scope of class field theory.

Consequence

When applied correctly, class field theory translates ideal-theoretic and idelic arithmetic into explicit descriptions of abelian extensions, predicts splitting behavior of primes, and serves as the foundation for modern reciprocity and Langlands-style generalizations.

Reversal

Neglecting class field theoretic structure leaves the behavior of abelian extensions opaque and makes the tracking of splitting and ramification across extensions ad hoc and less conceptual.

Boundary

Class field theory governs only abelian extensions; nonabelian extensions require additional frameworks. It presumes knowledge of ideles, local-global principles, and often analytic inputs for explicit constructions.

Semantic Tension

There is tension between concrete ideal-theoretic descriptions and the idelic, cohomological formulations: ideal class groups yield computable information, while idelic or cohomological language gives conceptual uniformity and better compatibility with generalizations.

Synthesis

Class field theory identifies abelian extensions of a base field with arithmetic quotients of its idele class group, converting algebraic extension data into explicit idelic and ideal-theoretic statements that control splitting, ramification, and reciprocity.