The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.
Statement.
Let formula_1 be a Galois extension of global fields and formula_2 stand for the idèle class group 
of formula_3. One of the statements of the Artin reciprocity law is that there is a canonical isomorphism called the global symbol map
 formula_4
where formula_5 denotes the abelianization of a group, and formula_6 is the Galois group of formula_3 over formula_8. The map formula_9 is defined by assembling the maps called the local Artin symbol, the local reciprocity map or the norm residue symbol
 formula_10
for different places formula_11 of formula_8. More precisely, formula_9 is given by the local maps formula_14 on the formula_11-component of an idèle class. The maps formula_14 are isomorphisms. This is the content of the "local reciprocity law", a main theorem of local class field theory.
Proof.
A cohomological proof of the global reciprocity law can be achieved by first establishing that
 formula_17
constitutes a class formation in the sense of Artin and Tate. Then one proves that
 formula_18
where formula_19 denote the Tate cohomology groups. Working out the cohomology groups establishes that formula_9 is an isomorphism.
Significance.
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 Hasse 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 the 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 global class field theory. It can be used to prove that Artin L-functions are meromorphic, and also to prove the Chebotarev density theorem.
Two years after the publication of his general reciprocity law in 1927, Artin rediscovered the transfer homomorphism of I. Schur and used the reciprocity law to translate the principalization problem for ideal classes of algebraic number fields into the group theoretic task of determining the kernels of transfers of finite non-abelian groups.
Finite extensions of global fields.
The definition of the Artin map for a finite abelian extension "L"/"K" of global fields (such as a finite abelian extension of formula_21) has a concrete description in terms of prime ideals and Frobenius elements.
If formula_22 is a prime of "K" then the decomposition groups of primes formula_23 above formula_22 are equal in Gal("L"/"K") since the latter group is abelian. If formula_22 is unramified in "L", then the decomposition group formula_26 is canonically isomorphic to the Galois group of the extension of residue fields formula_27 over formula_28. There is therefore a canonically defined Frobenius element in Gal("L"/"K") denoted by formula_29 or formula_30. If Δ denotes the relative discriminant of "L"/"K", the Artin symbol (or Artin map, or (global) reciprocity map) of "L"/"K" is defined on the group of prime-to-Δ fractional ideals, formula_31, by linearity:
formula_32
The Artin reciprocity law (or global reciprocity law) states that there is a modulus c of "K" such that the Artin map induces an isomorphism
formula_33
where "K"c,1 is the ray modulo c, N"L"/"K" is the norm map associated to "L"/"K" and formula_34 is the fractional ideals of "L" prime to c. Such a modulus c is called a defining modulus for "L"/"K". The smallest defining modulus is called the conductor of "L"/"K" and typically denoted formula_35
Examples.
Quadratic fields.
If formula_36 is a squarefree integer, formula_37 and formula_38, then formula_39 can be identified with {±1}. The discriminant Δ of "L" over formula_21 is "d" or 4"d" depending on whether "d" ≡ 1 (mod 4) or not. The Artin map is then defined on primes "p" that do not divide Δ by
formula_41
where formula_42 is the Kronecker symbol. More specifically, the conductor of formula_43 is the principal ideal (Δ) or (Δ)∞ according to whether Δ is positive or negative, and the Artin map on a prime-to-Δ ideal ("n") is given by the Kronecker symbol formula_44 This shows that a prime "p" is split or inert in "L" according to whether formula_42 is 1 or −1.
Cyclotomic fields.
Let "m" ; 1 be either an odd integer or a multiple of 4, let formula_46 be a primitive "m"th root of unity, and let formula_47 be the "m"th cyclotomic field. formula_39 can be identified with formula_49 by sending σ to "a"σ given by the rule
formula_50
The conductor of formula_43 is ("m")∞, and the Artin map on a prime-to-"m" ideal ("n") is simply "n" (mod "m") in formula_52
Relation to quadratic reciprocity.
Let "p" and formula_53 be distinct odd primes. For convenience, let formula_54 (which is always 1 (mod 4)). Then, quadratic reciprocity states that
formula_55
The relation between the quadratic and Artin reciprocity laws is given by studying the quadratic field formula_56 and the cyclotomic field formula_57 as follows. First, "F" is a subfield of "L", so if "H" = Gal("L"/"F") and formula_58 then formula_59 Since the latter has order 2, the subgroup "H" must be the group of squares in formula_60 A basic property of the Artin symbol says that for every prime-to-ℓ ideal ("n")
formula_61
When "n" = "p", this shows that formula_62 if and only if, "p" modulo ℓ is in "H", i.e. if and only if, "p" is a square modulo ℓ.
Statement in terms of "L"-functions.
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 formula_63 be an abelian Galois extension with Galois group "G". Then for any character formula_64 (i.e. one-dimensional complex representation of the group "G"), there exists a Hecke character formula_65 of "K" such that
formula_66
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.
The formulation of the Artin reciprocity law as an equality of "L"-functions allows formulation of a generalisation to "n"-dimensional representations, though a direct correspondence is still lacking.