Artin's reciprocity law implies a description of the abelianization of the absolute Galois group of a global field K which is based on the local–global principle and the use of the Frobenius elements. Together with the Takagi existence theorem, it is used to describe the abelian extensions of K in terms of arithmetic of K and to understand the behavior of the nonarchimedean places in them. Therefore, the Artin reciprocity law can be interpreted as one of the main theorems of the global class field theory.
called the local Artin symbol.
Let L⁄K be a Galois extension of global fields and CL stand for the idèle class group of L. The maps θv for different places v of K can be assembled into a single global symbol map. One of the statements of the Artin reciprocity law is that this results in the canonical isomorphism
A cohomological proof of the global reciprocity law can be achieved by first establishing that
constitutes a class formation in the sense of Artin and Tate. Then one proves that
where denote the Tate cohomology groups. Working out the cohomology groups establishes that θ is an isomorphism.
An alternative version of the reciprocity law, leading to the Langlands program, connects Artin L-functions associated to abelian extensions of a number field with Hecke L-functions associated to characters of the idèle class group.
A Hecke character (or Größencharakter) of a number field K is defined to be a quasicharacter of the idèle class group of K. Robert Langlands interpreted Hecke characters as automorphic forms on the reductive algebraic group GL(1) over the ring of adeles of K.
Let E⁄K be an abelian Galois extension with Galois group G. Then for any character σ: G → C× (i.e. one-dimensional complex representation of the group G), there exists a Hecke character χ of K such that
where the left hand side is the Artin L-function associated to the extension with character σ and the right hand side is the Hecke L-function associated with χ, Section 7.D of .