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The Artin reciprocity law, which was established by
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...
in a series of papers (1924; 1927; 1930), is a general theorem in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
that forms a central part of global
class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...
. The term "
reciprocity law In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f(x) with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irr ...
" 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 Hilbert's ninth problem, from the list of 23 Hilbert's problems (1900), asked to find the most general reciprocity law for the norm residues of ''k''-th order in a general algebraic number field, where ''k'' is a power of a prime. Progress ma ...
.


Statement

Let L/K be a
Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...
of
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function fi ...
s and C_L stand for the idèle class group of L. One of the statements of the Artin reciprocity law is that there is a canonical isomorphism called the global symbol mapNeukirch (1999) p.391 : \theta: C_K/ \to \operatorname(L/K)^, where \text denotes the
abelianization In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...
of a group. The map \theta is defined by assembling the maps called the local Artin symbol, the local reciprocity map or the
norm residue symbol In number theory, a symbol is any of many different generalizations of the Legendre symbol. This article describes the relations between these various generalizations. The symbols below are arranged roughly in order of the date they were introduce ...
Serre (1967) p.140Serre (1979) p.197 : \theta_v: K_v^/N_(L_v^) \to G^, for different places v of K. More precisely, \theta is given by the local maps \theta_v on the v-component of an idèle class. The maps \theta_v are isomorphisms. This is the content of the ''local reciprocity law'', a main theorem of
local class field theory In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite re ...
.


Proof

A cohomological proof of the global reciprocity law can be achieved by first establishing that : (\operatorname(K^/K),\varinjlim C_L) constitutes a
class formation In mathematics, a class formation is a topological group acting on a module satisfying certain conditions. Class formations were introduced by Emil Artin and John Tate to organize the various Galois groups and modules that appear in class field t ...
in the sense of Artin and Tate.Serre (1979) p.164 Then one proves that : \hat^(\operatorname(L/K), C_L) \simeq\hat^(\operatorname(L/K), \Z), where \hat^ denote the
Tate cohomology group In mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence. They were introduced by , and are used in class field theory. Defin ...
s. Working out the cohomology groups establishes that \theta is an isomorphism.


Significance

Artin's reciprocity law implies a description of the
abelianization In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...
of the absolute
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
of a
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function fi ...
''K'' which is based on the Hasse local–global principle and the use of the
Frobenius element In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism m ...
s. Together with the
Takagi existence theorem {{short description, Correspondence between finite abelian extensions and generalized ideal class groups In class field theory, the Takagi existence theorem states that for any number field ''K'' there is a one-to-one inclusion reversing correspond ...
, it is used to describe the
abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvabl ...
s 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-function In mathematics, an Artin ''L''-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group ''G''. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. T ...
s are
meromorphic In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
and for the proof of the
Chebotarev density theorem Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension ''K'' of the field \mathbb of rational numbers. Generally speaking, a prime integer will factor into several ideal ...
. 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 Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvabl ...
''L''/''K'' of
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function fi ...
s (such as a finite abelian extension of \Q) has a concrete description in terms of
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
s and
Frobenius element In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism m ...
s. If \mathfrak is a prime of ''K'' then the decomposition groups of primes \mathfrak above \mathfrak are equal in Gal(''L''/''K'') since the latter group is abelian. If \mathfrak is
unramified In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
in ''L'', then the decomposition group D_\mathfrak is canonically isomorphic to the Galois group of the extension of residue fields \mathcal_/\mathfrak over \mathcal_/\mathfrak. There is therefore a canonically defined Frobenius element in Gal(''L''/''K'') denoted by \mathrm_\mathfrak or \left(\frac\right). If Δ denotes the
relative discriminant In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the ( ring of integers of the) algebraic number field. More specifically, it is proportional to the squared vo ...
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, I_K^\Delta, by linearity: :\begin \left(\frac\right):I_K^\Delta \longrightarrow \operatorname(L/K)\\ \prod_^m\mathfrak_i^ \longmapsto \prod_^m\left(\frac\right)^ \end 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 :I_K^\mathbf/i(K_)\mathrm_(I_L^\mathbf)\overset\mathrm(L/K) where ''K''c,1 is the ray modulo c, N''L''/''K'' is the norm map associated to ''L''/''K'' and I_L^\mathbf 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 \mathfrak(L/K).


Examples


Quadratic fields

If d\neq1 is a
squarefree integer In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square- ...
, K=\Q, and L=\Q(\sqrt), then \operatorname(L/\Q) can be identified with . The discriminant Δ of ''L'' over \Q 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 :p\mapsto\left(\frac\right) where \left(\frac\right) is the
Kronecker symbol In number theory, the Kronecker symbol, written as \left(\frac an\right) or (a, n), is a generalization of the Jacobi symbol to all integers n. It was introduced by . Definition Let n be a non-zero integer, with prime factorization :n=u \cdot ...
. More specifically, the conductor of L/\Q 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 \left(\frac\right). This shows that a prime ''p'' is split or inert in ''L'' according to whether \left(\frac\right) is 1 or −1.


Cyclotomic fields

Let ''m'' > 1 be either an odd integer or a multiple of 4, let \zeta_m be a primitive ''m''th root of unity, and let L = \Q(\zeta_m) be the ''m''th
cyclotomic field In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of th ...
. \operatorname(L/\Q) can be identified with (\Z/m\Z)^ by sending σ to ''a''σ given by the rule :\sigma(\zeta_m)=\zeta_m^. The conductor of L/\Q is (''m'')∞, and the Artin map on a prime-to-''m'' ideal (''n'') is simply ''n'' (mod ''m'') in (\Z/m\Z)^.


Relation to quadratic reciprocity

Let ''p'' and \ell be distinct odd primes. For convenience, let \ell^* = (-1)^\ell (which is always 1 (mod 4)). Then, quadratic reciprocity states that :\left(\frac\right)=\left(\frac\right). The relation between the quadratic and Artin reciprocity laws is given by studying the quadratic field F=\Q(\sqrt) and the cyclotomic field L=\Q(\zeta_\ell) as follows. First, ''F'' is a subfield of ''L'', so if ''H'' = Gal(''L''/''F'') and G= \operatorname(L/\Q), then \operatorname(F/\Q) = G/H. Since the latter has order 2, the subgroup ''H'' must be the group of squares in (\Z/\ell\Z)^. A basic property of the Artin symbol says that for every prime-to-ℓ ideal (''n'') :\left(\frac\right)=\left(\frac\right)\pmod H. When ''n'' = ''p'', this shows that \left(\frac\right)=1 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 In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...
, connects
Artin L-function In mathematics, an Artin ''L''-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group ''G''. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. T ...
s associated to abelian extensions of a
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
with Hecke L-functions associated to characters of the idèle class group.James Milne
''Class Field Theory''
/ref> A
Hecke character In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of ''L''-functions larger than Dirichlet ''L''-functions, and a natural setting for the Dedekind zeta-functions and ce ...
(or Größencharakter) of a number field ''K'' is defined to be a quasicharacter of the idèle class group of ''K''.
Robert Langlands Robert Phelan Langlands, (; born October 6, 1936) is a Canadian mathematician. He is best known as the founder of the Langlands program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study o ...
interpreted Hecke characters as
automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
s on the
reductive algebraic group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...
''GL''(1) over the
ring of adeles Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film an ...
of ''K''.. Let E/K be an abelian Galois extension with
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
''G''. Then for any
character Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
\sigma: G \to \Complex^ (i.e. one-dimensional complex representation of the group ''G''), there exists a Hecke character \chi of ''K'' such that :L_^(\sigma, s) = L_^(\chi, s) 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.


Notes


References

*Emil Artin (1924) "Über eine neue Art von L-Reihen",
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (English: ''Reports from the Mathematical Seminar of the University of Hamburg'') is a peer-reviewed mathematics journal published by Springer Science+Business Media. It publishes articles on pure mathematics and is scientifically coordinated by ...
3: 89–108; ''Collected Papers'',
Addison Wesley Addison-Wesley is an American publisher of textbooks and computer literature. It is an imprint of Pearson PLC, a global publishing and education company. In addition to publishing books, Addison-Wesley also distributes its technical titles through ...
(1965), 105–124 *Emil Artin (1927) "Beweis des allgemeinen Reziprozitätsgesetzes", ''Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg'' 5: 353–363; ''Collected Papers'', 131–141 *Emil Artin (1930) "Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetzes", ''Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg'' 7: 46–51; ''Collected Papers'', 159–164 * * * * * * * * *{{citation , last=Tate , first=John , authorlink=John Tate (mathematician) , chapter=VII. Global class field theory , pages=162–203 , editor1-last=Cassels , editor1-first=J.W.S. , editor1-link=J. W. S. Cassels , editor2-last=Fröhlich , editor2-first=A. , editor2-link=Albrecht Fröhlich , title=Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union , location=London , publisher=Academic Press , year=1967 , zbl=0153.07403 Class field theory