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The Thirties

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The Story of Algebraic Numbers in the First Half of the 20th Century

Part of the book series: Springer Monographs in Mathematics ((SMM))

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Abstract

In this chapter we present first the progress in the study of the structure of number fields, the central subject being the existence of normal and normal integral bases, and then consider some additive questions, mainly on sums of squares. The next section concentrates on the simplification of the class-field theory by Hasse and Chevalley, and the following sections concern i.a. the class-number and class-group of quadratic fields, the question of the existence of Euclidean algorithm in fields, the distribution of algebraic integers on the complex plane and infinite extensions of number fields.

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Notes

  1. 1.

    For the theory of Prüfer domains see the book [1207] by Fontana, Huckaba and Papick.

  2. 2.

    Oscar Zariski (1899–1986), professor at the Johns Hopkins University and at Harvard. See [2996, 3218].

  3. 3.

    Pierre Samuel (1921–2009), professor in Clermont-Ferrand and at the Université Paris-Sud.

  4. 4.

    Israel Porush [Porusch-Mandel] (1907–1991), senior rabbi at the Great Synagoge in Sydney.

  5. 5.

    Cyrus Colton MacDuffee (1895–1961), professor at the Ohio State University and the University of Wisconsin.

  6. 6.

    Yasuo Akizuki (1902–1984), professor in Kyoto.

  7. 7.

    William Charles Waterhouse (1942–2016), professor at the Pennsylvania State University.

  8. 8.

    Tadashi Nakayama (1912–1964), professor in Nagoya. See [10].

  9. 9.

    Štefan Schwarz (1914–1996), professor in Bratislava. See [3464].

  10. 10.

    Carl Faith (1927–2014), professor at the Rutgers University.

  11. 11.

    Sam Perlis (1913–2009), professor at the Purdue University.

  12. 12.

    ‘Die zahlentheoretischen Anwendungen der Theorie ...hoffe ich in einer spätern Arbeit angeben zu können.” [“Number-theoretical applications of the theory ...I hope to present in a future paper.]”

  13. 13.

    Harald Nehrkorn (1910–2006), teacher in Schloss Bieberstein.

  14. 14.

    Paul Epstein (1871–1939), professor in Frankfurt.

  15. 15.

    László Rédei (1900–1980), professor in Szeged and Budapest. See [7, 2738].

  16. 16.

    Claude Chabauty (1910–1990), professor in Grenoble.

  17. 17.

    Pierre Liardet (1943–2014), professor at the Université de Provence. See [185, 4279]

  18. 18.

    Robert F. Coleman (1954–2014) professor at Berkeley. See [183].

  19. 19.

    Masatoshi Gündüz Ikeda (1926–2003), professor in Izmir and Ankara. See [2190].

  20. 20.

    Cahit Arf (1910–1997), professor in Istanbul. See [2631].

  21. 21.

    Joseph Henry Maclagen Wedderburn (1882–1948), professor in Princeton. See [4034].

  22. 22.

    For the first time ideles, under the name ‘éléments idéaux’ occur in a letter of Chevalley to Hasse [679] written in June 1935.

  23. 23.

    The definition of the restricted product in the general case occurs for the first time in the thesis of Braconnier [420, 421].

  24. 24.

    ...it permits in effect to avoid the somewhat delicate treatment of ’congruence groups’ with their several ’modules of definition’.

  25. 25.

    Epstein zeta-functions of quadratic forms were introduced in 1903 by Epstein [1112]. It has been found recently by Oswald and Steuding [3204] that these functions were earlier studied by A. Hurwitz.

  26. 26.

    Hans Arnold Heilbronn (1908–1975), professor in Bristol and Toronto. See [600].

  27. 27.

    Edward Hubert Linfoot (1905–1982), lecturer at the University of Bristol. See [281].

  28. 28.

    Nikolaĭ Grigorevič Čudakov (1904–1986), professor in Saratov, Moscow and Leningrad.

  29. 29.

    Theodor Estermann (1902–1991), Reader at the University College, London.

  30. 30.

    Emil Grosswald (1912–1989), professor at the Temple University.

  31. 31.

    A simple proof of Jacobi’s formula has been given in 2000 by Spearman and K.S. Williams [3866].

  32. 32.

    Hellmuth Kneser (1898–1973), professor in Greifswald and Tübingen, son of Adolf Kneser, father of Martin Kneser. See [4398].

  33. 33.

    Alexander Victor Oppenheim (1903–1997), professor in Singapore and Kuala Lumpur.

  34. 34.

    Chao Ko (1910–2002), professor at Sichuan University.

  35. 35.

    Charles Pisot (1910–1984), professor in Bordeaux and Paris. See [62].

  36. 36.

    Tirukannapuram Vijayaraghavan (1902–1955), professor at the Andhra University and the Ramanujan Institute for Mathematics in Madras. See [824].

  37. 37.

    Raphael Salem (1898–1963), professor at MIT, in Caen and Paris.

  38. 38.

    Antoni Szczepan Zygmund (1900–1992), professor in Wilno and Chicago. See [1166].

  39. 39.

    William Parry (1934–2006), professor in Warwick.

  40. 40.

    It has been noted in 1914 by Landau [2425] that Jensen’s formula in the case of polynomials occurs already in a paper of Jacobi [2006], published in 1827.

  41. 41.

    Johan Ludwig William Valdemar Jensen (1859–1925), worked in a telephone company. See [3129].

  42. 42.

    Weil pointed out that the first definition of the height for algebraic numbers appeared in the paper of Siegel [3777]. The name “height” (“Höhe” in German) has been introduced by Hasse in [1687].

  43. 43.

    Douglas Geoffrey Northcott (1916–2005), professor at Sheffield University. See [3751].

  44. 44.

    Tracy Augustus Pierce (1891–1945), professor at the University of Nebraska.

  45. 45.

    Robert Hermann Breusch (1907–1995), professor at the Amherst College.

  46. 46.

    John William Scott Cassels (1922–2015), professor in Cambridge.

  47. 47.

    David Geoffrey Cantor (1935–2012), professor at the University of Washington and the University of California at Los Angeles.

  48. 48.

    Ernst Gabor Straus (1922–1983), professor at the University of California at Los Angeles. See [1115, 1458].

  49. 49.

    Oswald Teichmüller (1913–1943), worked in Berlin. See [1719].

  50. 50.

    Jerzy Browkin (1934–2015), professor in Warsaw. See [3624].

  51. 51.

    Max Zorn (1906–1993), professor at Yale, University of California at Los Angeles and Indiana University.

  52. 52.

    John Todd (1911–2007), husband of Olga Taussky-Todd, professor at CalTech. See [215].

  53. 53.

    Karl Weierstrass (1815–1897), professor in Berlin. See [2411, 3325].

  54. 54.

    Henri Cartan (1904–2008), professor in Paris. See [3740].

  55. 55.

    Ernests Fogels (1910–1985), worked in Riga. See [2333].

  56. 56.

    See Ore [3186] for a review of it.

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  1. University of Wrocław, Wrocław, Poland

    Władysław Narkiewicz

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Narkiewicz, W. (2018). The Thirties. In: The Story of Algebraic Numbers in the First Half of the 20th Century. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-03754-3_5

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