A principal statement of class field theory giving the Artin map from the idele class group of a global field onto the abelianized Galois group of its maximal abelian extension, identifying abelian extensions with open subgroups of the idele class group and encoding Frobenius elements as images of ideles.

Principle

There is a canonical reciprocity homomorphism (the Artin map) whose kernel corresponds to norms from the maximal abelian extension; this map is functorial in finite abelian extensions and respects decomposition and inertia at primes.

Demonstration

For the rational field Q, the theorem specializes to Kronecker–Weber: every finite abelian extension of Q is contained in a cyclotomic field and the Artin map identifies residue classes modulo n with Frobenius automorphisms for primes not dividing n.

Misapplication

Using the Artin reciprocity statement to describe non-abelian extensions or expecting a bijection without passing to the appropriate quotient of the idele class group; also treating local and global reciprocity interchangeably without adjusting hypotheses.

Consequence

A complete classification of finite abelian extensions of global fields in terms of subgroups of ideles, explicit conductors and reciprocity laws for local and global norms, and a framework for interpreting abelian L-functions via characters of the idele class group.

Reversal

The inverse perspective—recovering the idele class group structure from a given abelian Galois group—fails in general without additional arithmetic input; non-abelian generalizations require different structures (e.g., Langlands program) rather than a direct reciprocity map.

Boundary

Applies to abelian extensions of global (and suitably modified local) fields; it does not provide a description for general non-abelian Galois groups, nor does it apply verbatim to arbitrary rings or non-global base fields.

Semantic Tension

Tension arises between the local reciprocity statements at primes (local class field theory) and the global Artin reciprocity law: the same Frobenius ideology manifests differently, and one must distinguish the global idele-class formulation from pointwise local reciprocity maps.

Synthesis

Artin reciprocity packages local Frobenius actions into a single global homomorphism from ideles to the abelianized Galois group, producing a bridge that classifies abelian extensions of global fields by arithmetic subgroups of the idele class group.