# Artin reciprocity law
The Artin reciprocity law is the central theorem of class field theory, the theory of abelian extensions of algebraic number fields: it identifies the [Galois group](https://www.edgepedia.org/galois-group) of any finite abelian extension with a quotient of a class group built from the base field itself, through a map that sends prime ideals to Frobenius automorphisms[1](https://ideas.repec.org/h/spr/sprchp/978-3-642-18908-1_8.html). [Emil Artin](https://www.edgepedia.org/emil-artin) (1898–1962) formulated the law in 1924 and proved it in 1927[1](https://ideas.repec.org/h/spr/sprchp/978-3-642-18908-1_8.html) • [2](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B25/pdf/B25_008.pdf). It generalizes the classical quadratic, cubic, and quartic reciprocity laws and, in modern terms, is the abelian case of the [Langlands program](https://www.edgepedia.org/langlands-program)[3](https://websites.math.leidenuniv.nl/stieltjes/archief/rep9899/node16.html).
| Key fact | Detail |
|---|---|
| Statement (global) | For a finite abelian extension L/K of number fields, the Artin map induces an isomorphism Cl_K^m / N_{L/K}(Cl_L^m) ≅ Gal(L/K), for a defining modulus m[4](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html) |
| Idelic form | The kernel of the global Artin homomorphism θ_K : C_K → Gal(K^ab/K) contains K×, giving C_K/N_{L/K}C_L ≅ Gal(L/K) for every finite abelian L/K[5](https://math.mit.edu/classes/18.785/2021fa/LectureNotes28.pdf) |
| Local form | For a finite extension K/k of local fields, the norm residue symbol induces k×/N_{K/k}K× ≅ Gal(K/k)^ab[6](https://www-personal.umd.umich.edu/~tiananw/Langlands%20learning%20notes.pdf) |
| Conjecture and proof | Conjectured by Artin in 1924; proved in "Beweis des allgemeinen Reziprozitätsgesetzes" (Abh. Hamburg, 1927)[2](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B25/pdf/B25_008.pdf) • [7](https://www.jstage.jst.go.jp/article/subutsukaishi1927/1/2/1_2_221/_article/-char/en) |
| Function fields | Hasse proved the reciprocity law for function fields in 1934, using cyclic algebras[8](https://math.stanford.edu/~conrad/252Page/handouts/cfthistory.pdf) |
| Consequence | The Principal Ideal Theorem: every ideal of K becomes principal in the Hilbert class field of K[9](https://www.jmilne.org/math/CourseNotes/CFTc.pdf) |
## The Artin map
For a prime ideal 𝔭 of a number field K that is unramified in a finite abelian extension L, there is a unique Frobenius automorphism (𝔭, L/K) in Gal(L/K), characterized by acting on residue fields as raising to the N𝔭-th power[10](https://www.jmilne.org/math/CourseNotes/CFT.pdf). Extending this assignment multiplicatively to fractional ideals coprime to the discriminant gives the global Artin map (or reciprocity map) Ψ_{L/K} : I_S → Gal(L/K)[10](https://www.jmilne.org/math/CourseNotes/CFT.pdf). For example, for the quadratic field K = Q(√m), the Artin map sends a prime p to the [Legendre symbol](https://www.edgepedia.org/legendre-symbol) (m/p), so quadratic reciprocity is literally a computation of this map[10](https://www.jmilne.org/math/CourseNotes/CFT.pdf).
The reciprocity law states that this map, which a priori depends on the extension L, is in fact controlled by K alone: it admits a modulus m (a formal product of an integral ideal and certain real primes), vanishes on the ray of elements congruent to 1 modulo m, and induces an isomorphism
\[ I^{\mathfrak{m}}_K / \bigl(i(K^{\mathfrak{m},1}) \cdot N_{L/K} I^{\mathfrak{m}}_L\bigr) \xrightarrow{\;\sim\;} \mathrm{Gal}(L/K). \]
The smallest such modulus is the conductor of L/K, and it is divisible exactly by the primes that ramify in L[4](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html) • [10](https://www.jmilne.org/math/CourseNotes/CFT.pdf). Together with the existence theorem, which attaches a class field to every congruence subgroup, this classifies all finite abelian extensions of K in terms of quotients of ray class groups[10](https://www.jmilne.org/math/CourseNotes/CFT.pdf) • [11](https://docs.magma-maths.org/GlobalFields/ClassFieldTheory/introduction.html).
## Idelic and local forms
Chevalley's introduction of ideles gave the law a formulation that works uniformly for number fields and function fields and extends to infinite abelian extensions[10](https://www.jmilne.org/math/CourseNotes/CFT.pdf). In the idelic version, the [idele class group](https://www.edgepedia.org/idele-class-group) C_K = I_K/K× replaces the ideal class group: the global Artin homomorphism θ_K : C_K → Gal(K^ab/K) has kernel containing K×, and for every finite abelian L/K it induces an isomorphism C_K/N_{L/K}C_L ≅ Gal(L/K)[5](https://math.mit.edu/classes/18.785/2021fa/LectureNotes28.pdf). For number fields θ_K is surjective but not injective; for global function fields it is injective with dense image[5](https://math.mit.edu/classes/18.785/2021fa/LectureNotes28.pdf).
The local analog, proved by Hasse as the first proof of local class field theory, states that for a finite extension K/k of local fields the norm residue symbol induces k×/N_{K/k}K× ≅ Gal(K/k)^ab[6](https://www-personal.umd.umich.edu/~tiananw/Langlands%20learning%20notes.pdf). The global and local maps are compatible: the global Artin map restricts at each place v to the local Artin map K_v× → Gal(L_w/K_v)[10](https://www.jmilne.org/math/CourseNotes/CFT.pdf).
## History
**Background.** Abelian extensions were studied through the nineteenth century by Kronecker, Weber, and Hilbert, building on the theory of cyclotomic fields begun by Gauss and carried forward by Kummer[1](https://ideas.repec.org/h/spr/sprchp/978-3-642-18908-1_8.html). Takagi (1920) proved the fundamental theorems of abelian class field theory as conjectured by Weber and Hilbert, establishing the class field correspondence and the isomorphism Gal(L/K) ≅ C_m/H, but without an arithmetic description of the isomorphism[1](https://ideas.repec.org/h/spr/sprchp/978-3-642-18908-1_8.html) • [10](https://www.jmilne.org/math/CourseNotes/CFT.pdf).
**Artin's contribution.** Much influenced by Takagi's theory, Artin formulated his general reciprocity law, conjecturally proposed in 1924, and showed that the power residue and norm residue reciprocity laws follow easily from it[2](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B25/pdf/B25_008.pdf). He published the proof in 1927, using [Otto Schreier](https://www.edgepedia.org/otto-schreier)'s analysis of Tschebotareff's (Chebotarev's) proof of the density theorem[2](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B25/pdf/B25_008.pdf). Takagi highly praised the result, which clarified the arithmetic nature of the isomorphism between a congruence ideal class group and the Galois group of its class field[2](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B25/pdf/B25_008.pdf).
**Later developments.** As an application, Artin formulated the Principal Ideal Theorem in 1930 in terms of the transfer (Verlagerung) from Gal(K̃/k)~ to its commutator subgroup, where K̃ is the [Hilbert class field](https://www.edgepedia.org/hilbert-class-field) of the Hilbert class field; the theorem itself, conjectured by Hilbert around 1900, had been proved by Furtwängler in 1930[2](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B25/pdf/B25_008.pdf) • [9](https://www.jmilne.org/math/CourseNotes/CFTc.pdf). In 1931 Artin introduced his L-functions, decomposing the [Dedekind zeta function](https://www.edgepedia.org/dedekind-zeta-function) ζ_K(s) of a [Galois extension](https://www.edgepedia.org/galois-extension) into a product of Artin L-functions and ζ_k(s)[2](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B25/pdf/B25_008.pdf). Hasse proved the reciprocity law for function fields in 1934[8](https://math.stanford.edu/~conrad/252Page/handouts/cfthistory.pdf), and Chevalley's purely algebraic idelic proof removed the analysis from the arguments and allowed a natural formulation for infinite abelian extensions[10](https://www.jmilne.org/math/CourseNotes/CFT.pdf). Artin and Tate then recast the whole theory cohomologically; their book *Class Field Theory*, based on a Princeton seminar and published by the American Mathematical Society in 1967, remains an original source for the global theory[12](https://books.google.com/books/about/Class_Field_Theory.html?id=8odbx9-9HBMC).
## Consequences and scope
The law yields the norm limitation theorem: for any finite extension L/K (not necessarily Galois), the norm group N_{L/K}C_L equals N_{M/K}C_M for M the maximal abelian subextension, so norm groups carry no information about the nonabelian part of an extension[13](https://kskedlaya.org/cft/sec_adelic-recip.html). Correspondingly, the splitting of primes in a nonabelian extension is not described by congruence conditions[9](https://www.jmilne.org/math/CourseNotes/CFTc.pdf).
Within class field theory, the reciprocity law underlies the Hilbert class field theory: the ray class field for the trivial modulus is the maximal unramified abelian extension, with Galois group isomorphic to the class group[14](https://math.mit.edu/classes/18.785/2015fa/LectureNotes21.pdf). The Chebotarev density theorem for abelian extensions follows from the existence of ray class fields together with the density of primes in ray classes[5](https://math.mit.edu/classes/18.785/2021fa/LectureNotes28.pdf).
Beyond class field theory, Artin reciprocity is the degree-one case of the Langlands program: Langlands's reciprocity conjecture, relating motivic L-functions to automorphic L-functions, was inspired by Artin's law, and the local-to-global compatibility of the Artin map lies at the heart of the extension of class field theory that the program envisions[3](https://websites.math.leidenuniv.nl/stieltjes/archief/rep9899/node16.html) • [13](https://kskedlaya.org/cft/sec_adelic-recip.html).
## References
1. [Günther Frei (2004). "On the History of the Artin Reciprocity Law in Abelian Extensions of Algebraic Number Fields." In *The Legacy of Niels Henrik Abel*, Springer, pp. 267–294.](https://ideas.repec.org/h/spr/sprchp/978-3-642-18908-1_8.html)
2. [Katsuya Miyake. "Takagi's Class Field Theory: From where? and to where?" *Algebraic Number Theory and Related Topics 2009*, RIMS Bessatsu B25.](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B25/pdf/B25_008.pdf)
3. [Langlands reciprocity, Stieltjes Institute archive, Leiden.](https://websites.math.leidenuniv.nl/stieltjes/archief/rep9899/node16.html)
4. [Romyar Sharifi. *Algebraic Number Theory*, Chapter 7: Global class field theory via ideals.](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html)
5. [MIT 18.785 Number Theory I, Fall 2021, Lecture 28 notes.](https://math.mit.edu/classes/18.785/2021fa/LectureNotes28.pdf)
6. [Langlands learning notes, University of Michigan–Dearborn.](https://www-personal.umd.umich.edu/~tiananw/Langlands%20learning%20notes.pdf)
7. [E. Artin. "Beweis des allgemeinen Reziprozitätsgesetzes." *Nippon Sugaku-Buturigakkwaishi* 1 (1927), 221–223.](https://www.jstage.jst.go.jp/article/subutsukaishi1927/1/2/1_2_221/_article/-char/en)
8. [Keith Conrad. "History of Class Field Theory."](https://math.stanford.edu/~conrad/252Page/handouts/cfthistory.pdf)
9. [J.S. Milne. *Class Field Theory* (comment version), Chapter V.](https://www.jmilne.org/math/CourseNotes/CFTc.pdf)
10. [J.S. Milne. *Class Field Theory* (v4.03, 2020).](https://www.jmilne.org/math/CourseNotes/CFT.pdf)
11. [Magma Handbook, Class Field Theory: Introduction.](https://docs.magma-maths.org/GlobalFields/ClassFieldTheory/introduction.html)
12. [Emil Artin and John Torrence Tate. *Class Field Theory*. American Mathematical Society, 1967.](https://books.google.com/books/about/Class_Field_Theory.html?id=8odbx9-9HBMC)
13. [Kiran Kedlaya. *Class Field Theory*, Section 6.4: The adelic reciprocity law and Artin reciprocity.](https://kskedlaya.org/cft/sec_adelic-recip.html)
14. [MIT 18.785 Number Theory I, Fall 2015, Lecture 21 notes.](https://math.mit.edu/classes/18.785/2015fa/LectureNotes21.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Algebraic number theory › Class field theory › Reciprocity laws*
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