{{Short description|Mathematical theorem}} 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.<ref>Helmut Hasse, ''History of Class Field Theory'', in ''Algebraic Number Theory'', edited by Cassels and Frölich, Academic Press, 1967, pp. 266&ndash;279</ref> 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 <math>L/K</math> be a Galois extension of global fields, <math>\text{Gal}(L/K)</math> be Galois group of this extension and <math>C_K, C_L</math> denote the idèle class groups of respectively <math>K</math> and <math>L</math>. Let: :<math>N_{L/K}: C_L \rightarrow C_K</math> denote idele norm map. The Artin reciprocity law states that there exist canonical isomorphism between quotient group and abelianization of Galois group:<ref name=N99391>Neukirch (1999) p.391</ref><ref>Jürgen Neukirch, ''Algebraische Zahlentheorie'', Springer, 1992, p. 408. In fact, a more precise version of the reciprocity law keeps track of the ramification.</ref> :<math> \theta: C_K/{N_{L/K}(C_L)} \to \operatorname{Gal}(L/K)^{\text{ab}}, </math> that is defined by assembling the local maps: :<math>\theta_v: K_v^{\times}/N_{L_v/K_v}(L_v^{\times}) \to \text{Gal}(L/K)^{\text{ab}},</math> for different places <math>v</math> of <math>K</math>.

=== Remarks === *The map <math>\theta</math> is called '''global symbol map''' or '''Artin symbol'''. *For each place <math>v</math> the local map <math>\theta_v</math> is called '''local Artin symbol''', '''local reciprocity map''' or '''norm residue symbol'''. *The local maps <math>\theta_v</math> are isomorphisms themselves, this is the content of local reciprocity law, key result in local class field theory.

===Proof===

A cohomological proof of the global reciprocity law can be achieved by first establishing that

: <math>(\operatorname{Gal}(K^{\text{sep}}/K),\varinjlim C_L)</math>

constitutes a class formation in the sense of Artin and Tate.<ref name=S79164>Serre (1979) p.164</ref> Then one proves that

: <math>\hat{H}^{0}(\operatorname{Gal}(L/K), C_L) \simeq\hat{H}^{-2}(\operatorname{Gal}(L/K), \Z),</math>

where <math>\hat{H}^{i}</math> denote the Tate cohomology groups. Working out the cohomology groups establishes that <math>\theta</math> is an isomorphism.

== Significance == {{See also|Quadratic reciprocity|Eisenstein reciprocity}} 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.<ref>Jürgen Neukirch, ''Algebraische Zahlentheorie'', Springer, 1992, Chapter VII</ref>

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.<ref>{{citation|title=Idealklassen in oberkörpern und allgemeines reziprozitätsgesetz|first=Emil|last=Artin|authorlink=Emil Artin|journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg|date=December 1929|volume=7|issue=1|pages=46–51|doi=10.1007/BF02941159}}.</ref>

==Finite extensions of global fields== (See [https://math.stackexchange.com/questions/4131855/frobenius-elements#:~:text=A%20Frobenius%20element%20for%20P,some%20%CF%84%E2%88%88KP math.stackexchange.com] for an explanation of some of the terms used here)

The definition of the Artin map for a finite abelian extension ''L''/''K'' of global fields (such as a finite abelian extension of <math>\Q</math>) has a concrete description in terms of prime ideals and Frobenius elements.

If <math>\mathfrak{p}</math> is a prime of ''K'' then the decomposition groups of primes <math>\mathfrak{P}</math> above <math>\mathfrak{p}</math> are equal in Gal(''L''/''K'') since the latter group is abelian. If <math>\mathfrak{p}</math> is unramified in ''L'', then the decomposition group <math>D_\mathfrak{p}</math> is canonically isomorphic to the Galois group of the extension of residue fields <math>\mathcal{O}_{L,\mathfrak{P}}/\mathfrak{P}</math> over <math>\mathcal{O}_{K,\mathfrak{p}}/\mathfrak{p}</math>. There is therefore a canonically defined Frobenius element in Gal(''L''/''K'') denoted by <math>\mathrm{Frob}_\mathfrak{p}</math> or <math>\left(\frac{L/K}{\mathfrak{p}}\right)</math>. 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, <math>I_K^\Delta</math>, by linearity:

:<math>\begin{cases} \left(\frac{L/K}{\cdot}\right):I_K^\Delta \longrightarrow \operatorname{Gal}(L/K)\\ \prod_{i=1}^m\mathfrak{p}_i^{n_i} \longmapsto \prod_{i=1}^m\left(\frac{L/K}{\mathfrak{p}_i}\right)^{n_i} \end{cases}</math>

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

:<math>I_K^\mathbf{c}/i(K_{\mathbf{c},1})\mathrm{N}_{L/K}(I_L^\mathbf{c})\overset{\sim}{\longrightarrow}\mathrm{Gal}(L/K)</math>

where ''K''<sub>'''c''',1</sub> is the ray modulo '''c''', N<sub>''L''/''K''</sub> is the norm map associated to ''L''/''K'' and <math>I_L^\mathbf{c}</math> 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 <math>\mathfrak{f}(L/K).</math>

===Examples===

====Quadratic fields==== If <math>d\neq1</math> is a squarefree integer, <math>K=\Q,</math> and <math>L=\Q(\sqrt{d})</math>, then <math>\operatorname{Gal}(L/\Q)</math> can be identified with {±1}. The discriminant Δ of ''L'' over <math>\Q</math> 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

:<math>p\mapsto\left(\frac{\Delta}{p}\right)</math>

where <math>\left(\frac{\Delta}{p}\right)</math> is the Kronecker symbol.<ref name="Lemmermeyer 2000 loc=§3.2">{{harvnb|Lemmermeyer|2000|loc=§3.2}}</ref> More specifically, the conductor of <math>L/\Q</math> is the principal ideal (Δ) or (Δ)∞ according to whether Δ is positive or negative,<ref>{{harvnb|Milne|2008|loc=example 3.11}}</ref> and the Artin map on a prime-to-Δ ideal (''n'') is given by the Kronecker symbol <math>\left(\frac{\Delta}{n}\right).</math> This shows that a prime ''p'' is split or inert in ''L'' according to whether <math>\left(\frac{\Delta}{p}\right)</math> is 1 or &minus;1.

====Cyclotomic fields==== Let ''m'' > 1 be either an odd integer or a multiple of 4, let <math>\zeta_m</math> be a primitive ''m''th root of unity, and let <math>L = \Q(\zeta_m)</math> be the ''m''th cyclotomic field. <math>\operatorname{Gal}(L/\Q)</math> can be identified with <math>(\Z/m\Z)^{\times}</math> by sending σ to ''a''<sub>σ</sub> given by the rule

:<math>\sigma(\zeta_m)=\zeta_m^{a_\sigma}.</math>

The conductor of <math>L/\Q</math> is (''m'')∞,<ref>{{harvnb|Milne|2008|loc=example 3.10}}</ref> and the Artin map on a prime-to-''m'' ideal (''n'') is simply ''n'' (mod ''m'') in <math>(\Z/m\Z)^{\times}.</math><ref>{{harvnb|Milne|2008|loc=example 3.2}}</ref>

===Relation to quadratic reciprocity=== Let ''p'' and <math>\ell</math> be distinct odd primes. For convenience, let <math>\ell^* = (-1)^{\frac{\ell-1}{2}}\ell</math> (which is always 1 (mod 4)). Then, quadratic reciprocity states that

:<math>\left(\frac{\ell^*}{p}\right)=\left(\frac{p}{\ell}\right).</math>

The relation between the quadratic and Artin reciprocity laws is given by studying the quadratic field <math>F=\Q(\sqrt{\ell^*})</math> and the cyclotomic field <math>L=\Q(\zeta_\ell)</math> as follows.<ref name="Lemmermeyer 2000 loc=§3.2"/> First, ''F'' is a subfield of ''L'', so if ''H'' = Gal(''L''/''F'') and <math>G= \operatorname{Gal}(L/\Q),</math> then <math>\operatorname{Gal}(F/\Q) = G/H.</math> Since the latter has order 2, the subgroup ''H'' must be the group of squares in <math>(\Z/\ell\Z)^{\times}.</math> A basic property of the Artin symbol says that for every prime-to-ℓ ideal (''n'')

:<math>\left(\frac{F/\Q}{(n)}\right)=\left(\frac{L/\Q}{(n)}\right)\pmod H.</math>

When ''n'' = ''p'', this shows that <math>\left(\frac{\ell^*}{p}\right)=1</math> 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.<ref name=milne>James Milne, [http://www.jmilne.org/math/CourseNotes/cft.html ''Class Field Theory'']</ref>

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''.<ref name=Gelbart>{{citation | last = Gelbart | first = Stephen S. | authorlink = Stephen Gelbart | location = Princeton, N.J. | mr = 0379375 | publisher = Princeton University Press | series = Annals of Mathematics Studies | title = Automorphic forms on adèle groups | volume = 83| year = 1975}}.</ref>

Let <math>E/K</math> be an abelian Galois extension with Galois group ''G''. Then for any character <math>\sigma: G \to \Complex^{\times}</math> (i.e. one-dimensional complex representation of the group ''G''), there exists a Hecke character <math>\chi</math> of ''K'' such that

:<math>L_{E/K}^{\mathrm{Artin}}(\sigma, s) = L_{K}^{\mathrm{Hecke}}(\chi, s)</math>

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.<ref name=Gelbart/>

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.

== See also == * List of eponymous laws

== Notes == <references/>

== References ==

*Emil Artin (1924) "Über eine neue Art von L-Reihen", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 3: 89–108; ''Collected Papers'', Addison Wesley (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| first=Günther| last=Frei|author-link=Günther Frei| contribution=On the history of the Artin reciprocity law in abelian extensions of algebraic number fields: how Artin was led to his reciprocity law| title=The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3--8, 2002| editor=Olav Arnfinn Laudal| editor2=Ragni Piene| pages=267–294 | publisher=Springer-Verlag | location=Berlin| year=2004 | mr=2077576| isbn=978-3-540-43826-7 | zbl=1065.11001}} *{{citation | first=Gerald | last=Janusz | title=Algebraic Number Fields | year=1973 | publisher=Academic Press | isbn=0-12-380250-4 | series=Pure and Applied Mathematics | volume=55 }} *{{Citation| last=Lang| first=Serge| author-link=Serge Lang| title=Algebraic number theory| edition=2| publisher=Springer-Verlag| year=1994| series= Graduate Texts in Mathematics| volume=110| place=New York| isbn=978-0-387-94225-4| mr=1282723}} *{{Citation| last=Lemmermeyer| first=Franz| author-link=Franz Lemmermeyer| title=Reciprocity laws: From Euler to Eisenstein| year=2000| publisher=Springer-Verlag| series=Springer Monographs in Mathematics| place=Berlin| mr=1761696 | zbl=0949.11002| isbn=978-3-540-66957-9}} *{{Citation| last=Milne| first=James| title=Class field theory| url=http://jmilne.org/math/CourseNotes/cft.html| edition=v4.0 | year=2008 | accessdate= 2010-02-22}} *{{citation | last=Neukirch | first=Jürgen | author-link=Jürgen Neukirch | title=Algebraic number theory | others=Translated from the German by Norbert Schappacher | series=Grundlehren der Mathematischen Wissenschaften | volume=322 | location=Berlin | publisher=Springer-Verlag | year=1999 | isbn=3-540-65399-6 | zbl=0956.11021 }} *{{citation | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=Local Fields |translator-link=Marvin Greenberg|translator-first1= Marvin Jay |translator-last1=Greenberg | series=Graduate Texts in Mathematics | volume=67 | location=New York, Heidelberg, Berlin | publisher=Springer-Verlag | year=1979 | isbn=3-540-90424-7 | zbl=0423.12016 }} *{{citation | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | chapter=VI. Local class field theory | pages=128–161 | 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 }} *{{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 }}

Category:Class field theory