Artin Reciprocity

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.

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.