On the History of the Artin Reciprocity Law in Abelian Extensions of Algebraic Number Fields: How Artin was Led to his Reciprocity Law

  • Günther Frei


Artin’s Reciprocity Law (1923/26) for Abelian extensions of algebraic number fields is the central theorem of Class Field Theory which, by the Theory of Takagi (1920) [64], is the Theory of Abelian Extensions of Algebraic Number Fields. Abelian extensions of algebraic number fields have been studied extensively in the second half of the 19-th century,in particular by Kronecker(1853,1856,1877,1882), Weber (1886/87, 1897/98) and Hilbert (1896, 1897, 1898) who laid the foundations and discovered many fundamental properties of the Class Fields. These discoveries were made possible by the thorough study of a particular kind of Abelian extensions, namely the study of cyclotomic fields initiated by Gauss (1801) [24, Sect. 7] and carried further by Kummer (1847-1874). The term Abelian in connection with algebraic extensions (or, at that time, algebraic equations) was coined by Kronecker in 1853, first related to algebraic equations as ‘Abelian equation.’ 1 By this Kronecker referred to polynomials with cyclic Galois group. However, Kronecker was already then aware of the more general polynomials with Abelian (in the modern sense) Galois group, since he is referring on page 6 (in [45], Werke, Bd. 4) to the fundamental treatise of Abel of 1829 which appeared in Volume 4 of Crelle’s Journal [1], just two months before Abel’s death, and where Abel states explicitly the condition for commutativity θθ 1x =θ1θx in his assertion that a polynomial with Abelian Galois group is solvable (with radicals). Abel says there on page 479 of [1]: “En général j’ai démontré le théorème suivant: ‘Si les racines d’une équation d’un degré quelconque sont liées entre elles de telle sorte, que toutes ces racines puissent être exprimées rationnellement au moyen de l’une d’elles, que nous d’signerons par x; si de plus, en désignant par θx, θ1 x deux autres racines quelconques, on a θ1 x = θθ 1 6x, 1’ equation dont il s’agit sera toujours résoluble algébriquement.’ ” (In general I have demonstrated the following theorem: If the roots of an equation of any degree are related to each other in such a way that all these roots can be expressed rationally by means of one of them, which will be denoted by x; if, in addition, we have θθ1x = θ1θx, where we denote by θx, θ1 x any two other roots, then the equation in question is algebraically solvable [that is, solvable by radicals].) Later in 1877 [48] Kronecker used the term ‘Abelian equation’in the larger sense to mean a polynomial with Abelian Galois group in our modern sense. He says there on page 66 (in [48], Werke, Bd. 4) that he now calls ‘Abelian equation’ an equation having the property that all its roots x are rational functions of any one of them and if θ 1 and θ2 are two of these functions then θ1θ2x = θ2θ1x. He then calls ‘simple Abelian equation’ the equation he treated in 1853, namely the equation with cyclic Galois group of prime order. That he had already used the term ‘Abelian equation’ in the paper of 1853 in the special case of cyclic equations is justified by Kronecker by the fact that the ‘Abelian equation’ can be reduced to the ‘simple Abelian equation,’ or as Kronecker says on page 69 (in [48], Werke, Bd. 4), that every root of any Abelian equation is a rational function of roots of simple Abelian equations.2 This justification is repeated in his paper of 1882 on the composition of Abelian equations.3 In the paper of 1853 Kronecker also states the famous theorem that any root of a (simple) Abelian equation [that is a cyclic equation] with integral rational coefficients can be represented by roots of unity, that is, is contained in a cyclotomic field over the rational number field.4 The same theorem is stated by Kronecker in his paper of 1877 in the case of the general Abelian equation with integral rational coefficients.5 It was later proved by Weber (1886) [67] and more simply by Hilbert (1896) [41]. In both papers Kronecker also stated his Jugendtraum for an analogous theorem for Abelian equations with coefficients in a quadratic imaginary number field. This theorem was proved partially by Weber (1908) [69] and Fueter (1914) [23] and completely by Takagi (1920) [64]. Both theorems have plaid a crucial rôle in the history of class field theory.6


Prime Ideal Galois Group Class Number Irreducible Polynomial Riemann Hypothesis 
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  1. [1]
    Abel, Niels Henrik: Mémoire sur une classe particulière d’équations résolubles algébriquement. J. Reine Angew. Math. 4 (1829). Oeuvres completes, Nouvelle édition par MM. L. Sylow et S. Lie, tome I, XXV. Christiania, 1881, pp. 478–507.Google Scholar
  2. [2]
    Artin, Emil: Uber die Zetafunktionen gewisser algebraischer Zahlkörper. Math. Ann. 89 (1923), 147–156.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Artin, Emil: Über eine neue Art von L-Reihen. Abh. Math. Sem. Hamburg 3 (1923), 89–108.zbMATHCrossRefGoogle Scholar
  4. [4]
    Artin, Emil: Quadratische Körper im Gebiete der höheren Kongruenzen I, II. Math. Zeitschrift 19 (1924), 153–246.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Artin, Emil: Beweis des allgemeinen Reziprozitatsgesetzes. Abh. Math. Sem. Hamburg 5 (1927), 353–363.zbMATHCrossRefGoogle Scholar
  6. [6]
    Artin, Emil: Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren, Abh. Math. Sem. Hamburg 8 (1930), pp. 292–306; Coll. Papers, pp. 165-179.zbMATHCrossRefGoogle Scholar
  7. [7]
    Artin, Emil: The Collected Papers of Emil Artin. Edited by Serge Lang and John T. Tate. Addison-Wesley, 1965.Google Scholar
  8. [8]
    Brauer, Richard: On Artin’ s L-series with general group characters. Ann. Math. (2) 48 (1947), pp. 502–514.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Cassels, J. W. S. and Frohlich, A.: Algebraic Number Theory. Academic Press, London, 1967.zbMATHGoogle Scholar
  10. [10]
    Dedekind, Richard: Abriß einer Theorie der höheren Kongruenzen in bezug auf einen reellen Primzahl-Modulus. J. Reine Angew. Math. 54 (1857), 1–26.zbMATHCrossRefGoogle Scholar
  11. [11]
    Dedekind, Richard: Uber den Zusammenhang zwischen der Theorie der Ideale und der Theorie der hoheren Kongruenzen. Abh. Königl. Ges. Wiss. Göttingen 23 (1878), 1–23; Ges. math. Werke, Bd. 1, XV, pp. 202-232.Google Scholar
  12. [12]
    Dedekind, Richard: Über die Anzahl der Idealklassen in reinen kubischen Körpern. J. Reine Angew. Math. 121 (1900), 40–123; Werke, Bd. 2, XXIX, pp. 148-235.Google Scholar
  13. [13]
    Dedekind, Richard: Gesammelte mathematische Werke. Herausgegeben von Robert Fricke, Emmy Noether, Öystein Ore. Erster Band, Braunschweig 1930; Zweiter Band, Braunschweig 1931; Dritter Band, Braunschweig 1932.Google Scholar
  14. [14]
    Dirichlet, Lejeune; Dedekind, Richard: Vorlesungen uber Zahlentheorie. Erste Auflage. Vieweg, Braunschweig, 1869.Google Scholar
  15. [15]
    Dirichlet, Lejeune; Dedekind, Richard: Vorlesungen uber Zahlentheorie. Vierte Auflage. Vieweg, Braunschweig, 1894.Google Scholar
  16. [16]
    Dedekind, Richard and Weber, Heinrich: Theorie der algebraischen Funktionen einer Veränderlichen. J. Reine Angew. Math. 92 (1882), pp. 181–290; Dedekind, Ges. math. Werke, Bd. 1, XVIII, pp. 238-350.zbMATHGoogle Scholar
  17. [17]
    Frei, Günther: Heinrich Weber and the Emergence of Class Field Theory. In: The History of Modern Mathematics, edited by David Rowe and John McCleary (2 Vol.), Academic Press, Boston (1989), Vol. 1, pp. 425–450.Google Scholar
  18. [18]
    Frei, Günther: The Reciprocity Law from Euler to Eisenstein. In: The Intersection of History and Mathematics (Editors: Sasaki Ch., Sugiura M., Dauben J.W.). Birkhäuser, Basel, 1994, pp. 67–88.CrossRefGoogle Scholar
  19. [19]
    Frei, Günther: How Hasse was led to the Theory of Quadratic Forms, the Local-Global Principle, the Theory of the Norm Residue Symbol, the Reciprocity Laws, and to Class Field Theory. In: Class Field Theory — Its Centenary and Prospect, Edited by Katsuya Miyake. Advanced Studies in Pure Mathematics 30. Mathematical Society of Japan. Tokyo, 2001, pp. 31–62.Google Scholar
  20. [20]
    Frei, Günther: On the Development of the Theory of Function Fields over a Finite Field from Gauss to Dedekind and Artin. Preprint 2001. 62 pages.Google Scholar
  21. [21]
    Frei, Günther: On the History of the Reciprocity Laws. Eleven letters from Artin to Hasse in the years 1923-1927. Preprint 2002.Google Scholar
  22. [22]
    Frei, Günther: Gauss’ unpublished Section Eight of the Disquisitiones arithmeticae: The Beginning of the Theory of Function Fields over a Finite Field. To appear in: The shaping of arithmetic two hundred years of number theory after C.F. Gauß’s Disquisitiones Arithmeticae (edited by C. Goldstein, N. Schappacher, J. Schwermer). Springer-Verlag, Heidelberg, 2003.Google Scholar
  23. [23]
    Fueter, Rudolf: Abelsche Gleichungen in quadratisch-imaginären Zahlkörpern. Math. Ann. 75 (1914), 177–255.MathSciNetCrossRefGoogle Scholar
  24. [24]
    Gauss, Carl Friedrich: Disquisitiones arithmeticae. Werke, Erster Band, Gottingen, 1863. (German translation in [56]; English translation in [27], Second corrected printing, Springer, Heidelberg, 1986.)Google Scholar
  25. [25]
    Gauss, Carl Friedrich: Werke. Zweiter Band. Göttingen, 1863.Google Scholar
  26. [26]
    Gauss, Carl Friedrich: Werke. Zweiter Band, Zweiter Abdruck. Göttingen, 1876.Google Scholar
  27. [27]
    Carl Friedrich Gauss: Disquisitiones arithmeticae. Translated by Arthur A. Clarke. Yale University Press, New Haven, 1966.zbMATHGoogle Scholar
  28. [28]
    Gauss, Carl Friedrich: Mathematisches Tagebuch 1796-1814. Mit einer historischen Einführung von Kurt-R. Biermann. Ins Deutsche übertragen von Elisabeth Schuhmann. Durchgesehen und mit Anmerkungen versehen von Hans Wußing und Olaf Neumann. 4. Auflage. Ostwalds Klassiker der exakten Wissenschaften 256. Akademische Verlagsgesellschaft Geest & Portig, Leipzig, 1985.Google Scholar
  29. [29]
    Gray, J. J.: A commentary on Gauss’s mathematical diary, 1796-1814, with an English translation. Expo. Math. 2 (1986), pp. 97–130.Google Scholar
  30. [30]
    Hasse, Helmut: History of Class Field Theory. In: Algebraic Number Theory, Edited by J. W. S. Cassels and A. Fröhlich. Academic Press, London, 1967, pp. 266–279.Google Scholar
  31. [31]
    Hecke, Erich: Über die Zetafunktion beliebiger algebraischer Zahlkörper. Nachr. K. Ges. Wiss. Göttingen, 1917, pp. 77–89; Werke, 7, pp. 159-171.Google Scholar
  32. [32]
    Hecke, Erich: Uber eine neue Anwendung der Zetafunktionen auf die Arithmetik der Zahlkörper. Nachr. K. Ges. Wiss. Göttingen, 1917, pp. 90–95; Werke, 8, pp. 172-177.Google Scholar
  33. [33]
    Hecke, Erich: Über die L-Funktionen und den Dirichletschen Primzahlsatz für einen beliebigen Zahlkörper. Nachr. K. Ges. Wiss. Göttingen, 1917, pp. 299–318; Werke, 9, pp. 178-197.Google Scholar
  34. [34]
    Hecke, Erich: Über die Kroneckersche Grenzformel für reelle, quadratische Körper und die Klassenzahl relativ-Abelscher Korper. Verh. Naturf. Ges. Basel 28 (1917), pp. 363–372; Werke, 10, pp. 198-207.Google Scholar
  35. [35]
    Herglotz, Gustav: Über das quadratische Reziprozitätsgesetz in imaginären quadratischen Zahlkörpern. Leipziger Ber. 73 (1921), pp. 303–310; Ges. Schriften, pp. 396-403.Google Scholar
  36. [36]
    Herglotz, Gustav: Zur letzten Eintragung im Gaußschen Tagebuch. Leipziger Ber. 73 (1921), pp. 271–276; Ges. Schriften, pp. 415-420.Google Scholar
  37. [37]
    Herglotz, Gustav: Über einen Dirichletschen Satz. Math. Zeitschrift 12 (1922), pp. 255–261; Ges. Schriften, pp. 429-435.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    Herglotz, Gustav: Über die Entwicklungskoeffizienten der Weierstraßschen ℘-Funktion. Leipziger Ber. 74 (1922), pp. 269–289; Ges. Schriften, pp. 436-456.Google Scholar
  39. [39]
    Herglotz, Gustav: Über die Kroneckersche Grenzformel für reelle quadratische Körper, I, II. Leipziger Ber. 75 (1923), pp. 3–14 and pp. 31-37; Ges. Schriften, pp. 466-484.Google Scholar
  40. [40]
    Hilbert, David: Über den Dirichletschen biquadratischen Zahlkörper. Math. Ann. 45 (1894), 309–340; Ges. Abh., Bd. I, 5, pp. 24-52.MathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    Hilbert, David: Ein neuer Beweis des Kroneckerschen Fundamentalsatzes über Abelsche Zahlkörper. Nachr. Ges. Wiss. Göttingen, 1896, pp. 29–39; Ges. Abh., Bd. I, 6, pp. 53-62.Google Scholar
  42. [42]
    Hilbert, David: Die Theorie der algebraischen Zahlkörper, J.-bericht DMV 4 (1897), pp. 175–546; Ges. Abh., Bd. I, 7, pp. 63-363.Google Scholar
  43. [43]
    Hilbert, David: Über die Theorie des relativquadratischen Zahlkörpers. Math. Ann. 51 (1899), 1–127; Ges. Abh., Bd. I, 9, pp. 370-482.zbMATHCrossRefGoogle Scholar
  44. [44]
    Kornblum, Heinrich: Über die Primfunktionen in einer arithmetischen Progression. Math. Zeitschrift 5 (1919), 100–111.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [45]
    Kronecker, Leopold: Über die algebraisch auflösbaren Gleichungen (I. Abhandlung). Monatsberichte der Königl. Preuss. Akademie der Wiss. zu Berlin, 1853, pp. 365–374; Werke, Bd. 4,I, pp. 1-11.Google Scholar
  46. [46]
    Kronecker, Leopold: Über die algebraisch auflösbaren Gleichungen (II. Abhandlung). Monatsberichte der Königl. Preuss. Akademie der Wiss. zu Berlin, 1856, pp. 203–215; Werke, Bd. 4, IV, pp. 25-37.Google Scholar
  47. [47]
    Kronecker, Leopold: Über den Gebrauch der Dirichlet’schen Methoden in der Theorie der quadratischen Formen. Monatsberichte der Königl. Preuss. Akademie der Wiss. zu Berlin, 1864, pp. 285–303; Werke, Bd. 4, XXII, pp. 227-244.Google Scholar
  48. [48]
    Kronecker, Leopold: Über Abelsche Gleichungen. Monatsberichte der Königl. Preuss. Akademie der Wiss. zu Berlin, 1877, pp. 845–851; Werke, Bd. 4, IX, pp. 63-71.Google Scholar
  49. [49]
    Kronecker, Leopold: Grundzüge einer arithmetischen Theorie der algebraischen Grössen. J. Reine Angew. Math. 92 (1881), 1–122; Werke, Bd. 2, XI, pp. 237-387.Google Scholar
  50. [50]
    Kronecker, Leopold: Die Composition Abelscher Gleichungen. Sitzungsberichte der Königl. Preuss. Akademie der Wiss. zu Berlin, 1882, pp. 1059–1064; Werke, Bd. 4, XII, pp. 113-121.Google Scholar
  51. [51]
    Kronecker, Leopold: Werke. Herausgegeben von K. Hensel. Bd. 1 (1895), Bd. 2 (1897), Bd. 3.1 (1899), Bd. 3.2 (1931), Bd. 4 (1929), Bd. 5 (1930).Google Scholar
  52. [52]
    Kuhne, H.: Eine Wechselbeziehung zwischen Functionen mehrerer Unbestimmten, die zu Reciprocitätsgesetzen führt. J. Reine Angew. Math. 124 (1902), 121–133.Google Scholar
  53. [53]
    Kuhne, H.: Angenaherte Auflösung von Congruenzen nach Primmodulsystemen in Zusammenhang mit den Einheiten gewisser Körper, J. Reine Angew. Math. 126 (1903), 102–115.Google Scholar
  54. [54]
    Kummer, Ernst Eduard: Zwei neue Beweise der allgemeinen Reciprocitätsgesetze unter den Resten und Nichtresten der Potenzen, deren Grad eine Primzahl ist. J. Reine Angew. Math. 100 (1887), 10–50; Reprint of Abhandl. der königl. Akademie der Wiss. zu Berlin 1861, pp. 81-122; Collected Papers, Vol. I, pp. 842-882.Google Scholar
  55. [55]
    Landau, Edmund: Handbuch der Lehre von der Verteilung der Primzahlen. Göttingen, 1909.Google Scholar
  56. [56]
    Maser, H.: Carl Friedrich Gauss’ Untersuchungen uber höhere Arithmetik. Julius Springer, Berlin, 1889.Google Scholar
  57. [57]
    Miyake, Katsuya: The Establishment of the Takagi—Artin Class Field Theory. In: The Intersection of History and Mathematics, Edited by S. Sasaki, M. Sugiura, J. W. Dauben. Birkhauser, Basel, 1994, pp. 109–128.Google Scholar
  58. [58]
    Moore, E. H.: Mathematical Papers Read at the International Mathematics Congress. Chicago, 1893; Macmillan, New York, 1896, pp. 208–226.Google Scholar
  59. [59]
    Pieper, Herbert: Variationen über ein zahlentheoretisches Thema von Carl Friedrich Gauss. Birkhäuser Verlag, Basel, 1978.Google Scholar
  60. [60]
    Roquette, Peter: On the history of Artin’s L-functions and conductors. Mitt. Math. Ges. Hamburg 19 (2000), 5–50.Google Scholar
  61. [61]
    Roquette, Peter: Class Field Theory in Characteristic p, its Origin and Development. In: Class Field Theory —Its Centenary and Prospect, edited by K. Miyake, Advanced Studies in Pure Mathematics 30, Tokyo (2001), pp. 549–631.Google Scholar
  62. [62]
    Schmidt, F. K.: Zur Zahlentheorie in Körpern der Charakteristik p. Vorläufige Mitteilung. Sitz.-Ber. phys. med. Soz. Erlangen 58/59 (1926/27), 159–172.Google Scholar
  63. [63]
    Schmidt, F. K.: Analytische Zahlentheorie in Körpern der Charakteristik p. Math. Zeitschr. 33 (1931), 1–32.CrossRefGoogle Scholar
  64. [64]
    Takagi, Teiji: Über eine Theorie des relativ Abel’schen Zahlkörpers. Journ. of Coll. of Science, Univ. of Tokyo 41, Art 9, 1920, 1–133.Google Scholar
  65. [65]
    Tschebotareff (Tchebotarev), N.: Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören. Math. Ann. 95 (1926), 191–228.MathSciNetCrossRefGoogle Scholar
  66. [66]
    Ullrich, Peter: Emil Artin’s unveröffentlichte Verallgemeinerung seiner Dissertation. Mitt. Math. Ges. Hamburg XIX (2000), 173–194.MathSciNetGoogle Scholar
  67. [67]
    Weber, Heinrich: Theorie der Abelschen Zahlkörper I, II. Acta Math. 8 (1886), pp. 193–263; 9 (1887), pp. 105-130.MathSciNetzbMATHCrossRefGoogle Scholar
  68. [68]
    Weber, Heinrich: Über Zahlengruppen in algebraischen Zahlkörpern I, II, III. Math. Ann. 48 (1897), pp. 433–473; 49 (1897), pp. 83-100; 50 (1898), pp. 1-26.MathSciNetCrossRefGoogle Scholar
  69. [69]
    Weber, Heinrich: Lehrbuch der Algebra. Dritter Band. Vieweg, Braunschweig, 1908.Google Scholar

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