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Abstract

The central themes of the fourth chapter are the creation of the modern ideal theory by Emmy Noether, and the proofs of the fundamental theorems of the class-field theory by Takagi and Artin. Also various other questions were considered at that time, for example the first results in the additive theory of algebraic numbers obtained by Rademacher.

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Notes

  1. 1.

    Alfred Pringsheim (1850–1941), professor in München, father-in-law of the writer Thomas Mann. See [3257].

  2. 2.

    Eugéne Cahen (1865–1941), teacher in Paris.

  3. 3.

    Lars Edward Phragmén (1863–1937), professor in Stockholm. See [568].

  4. 4.

    Hermann Kinkelin (1832–1913), professor in Basel.

  5. 5.

    Hermann Weyl (1885–1955), professor at ETH in Zürich, Göttingen and Princeton. See [686, 3068, 3109].

  6. 6.

    Arnold Walfisz (1892–1962), professor in Tbilisi. See [2269].

  7. 7.

    Takayoshi Mitsui (1929–1997), professor at the Gakushuin University in Toshima.

  8. 8.

    Nikolaĭ Mihailovič Korobov (1917–2004), professor in Moscow. See [6].

  9. 9.

    Hans-Egon Richert (1924–1993), professor in Marburg and Ulm.

  10. 10.

    Włodzimierz Staś (1925–2011), professor in Poznań.

  11. 11.

    Jacques Salomon Hadamard (1865–1963), professor in Bordeaux and Paris. See [2586, 2794].

  12. 12.

    Pafnutiĭ Lvovič Čebyšev (1821–1894), professor in St. Petersburg. See [3341].

  13. 13.

    Oskar Perron (1880–1975), professor in Tübingen, Heidelberg and München. See [1230, 1758].

  14. 14.

    Wacław Sierpiński (1882–1969), professor in Lwów and Warsaw. See [3617].

  15. 15.

    Kanakanahalli Ramachandra (1933–2011), professor at the Tata Institute and the National Institute of Advanced Studies in Bangalore. See [3001].

  16. 16.

    Bernhard Riemann (1826–1866), professor in Göttingen. See [2487].

  17. 17.

    The actual writing of the functional equation for \(\zeta _k(s)\) is, as far I know, not yet done but presents no principal difficulties”.

  18. 18.

    Jürgen Neukirch (1937–1997), professor in Regensburg.

  19. 19.

    I did not perform the calculation”.

  20. 20.

    Erhard Schmidt (1876–1959), professor in Zürich, Erlangen, Breslau and Berlin. See [948, 3494].

  21. 21.

    In [795] his name is written “Tchudakoff”.

  22. 22.

    Paul Turán (1910–1976), professor at the Budapest University. See [1114, 1593].

  23. 23.

    Thomas Haakon Gronwall (1877–1932), worked in Princeton and New York. See [1850].

  24. 24.

    Characters according to units.

  25. 25.

    Characters of magnitude.

  26. 26.

    Stanisław Knapowski (1931–1967), professor in Poznań and Miami. See [4083].

  27. 27.

    Hans Rademacher (1892–1969), professor in Breslau and the University of Pennsylvania. See [79].

  28. 28.

    Jonas Kubilius (1921–2011), professor in Vilnius. See [2737].

  29. 29.

    Kȩstutis Antanas Bulota (1929–1990), professor in Vilnius.

  30. 30.

    Mindaugas Maknys [Maknis] (1944–1992), professor in Vilnius.

  31. 31.

    Jonušas Adolfas Urbelis [Urbjalis] (1927–2015), professor in Kaunas.

  32. 32.

    John Edensor Littlewood (1885–1977), professor in Cambridge. See [520].

  33. 33.

    Yurij Vladimirovič Linnik (1915–1972), professor in Leningrad. See [1947, 2730].

  34. 34.

    Paul Trevier Bateman (1919–2012), professor at the University of Illinois in Urbana-Champaign. See [12].

  35. 35.

    Claude Chevalley (1909–1984), professor at the Columbia University and in Paris. See [944, 2004].

  36. 36.

    Karl Dörge (1899–1975), professor in Köln.

  37. 37.

    Friedrich Karl Schmidt (1901–1977), professor in Jena, Münster and Heidelberg. See [2379].

  38. 38.

    The height of a polynomial \(P(X)=\sum _{j=0}^{n}a_jX^j\in \mathbf{Z}[X]\) equals \(\max _i|a_i|\).

  39. 39.

    Raymond Joseph Garver (1901–1935), professor at the University of California at Los Angeles. See [4396].

  40. 40.

    Wolfgang Gröbner (1899–1980), professor in Innsbruck.

  41. 41.

    Abraham Adrian Albert (1905–1972), professor at the University of Chicago. See [2017, 2120].

  42. 42.

    Ernst Sigismund Fischer (1899–1983), professor in Erlangen and Köln.

  43. 43.

    John Arthur Todd (1908–1994), Reader in Cambridge. See [150].

  44. 44.

    Legh Wilber Reid (1867–1961), professor at Haverford College.

  45. 45.

    For ideles see Sect. 5.2.2.

  46. 46.

    If a group G has a non-trivial subgroup H whose intersections with its conjugates are trivial, then G is called a Frobenius group.

  47. 47.

    Carl Ferdinand Degen (1766–1825), professor in Copenhagen.

  48. 48.

    Edward Everett Whitford (1865–1946), instructor at the College of the City of New York.

  49. 49.

    Derrick Norman Lehmer (1867–1938), father of Derrick Henry Lehmer, professor at Berkeley. See [3347].

  50. 50.

    Albert Châtelet (1883–1960), professor in Lille, Caen and at the Sorbonne.

  51. 51.

    Henry Cabourn Pocklington (1870–1952), fellow of the Royal Society and St. John’s College at Cambridge. He worked as a teacher of physics in Leeds. See [3520].

  52. 52.

    Lowell Schoenfeld (1920–2002), professor at the Pennsylvania State University and in Buffalo.

  53. 53.

    Fritz Gassmann (1899–1990), professor at the ETH in Zürich.

  54. 54.

    Donald John Lewis (1926–2015), professor at Ann Arbor.

  55. 55.

    Allan Joseph Champneys Cunningham (1842–1928), Fellow of King’s College, London. See [4371].

  56. 56.

    Alexander Aigner (1909–1988), professor in Graz.

  57. 57.

    Albert Leon Whiteman (1915–1995), professor at the University of Southern California. See [1473].

  58. 58.

    Boris Alekseevič Venkov [Wenkov] (1900–1962), father of B.B. Venkov, professor in Leningrad.

  59. 59.

    George Szekeres (1911–2005), professor in Adelaide and the University of New South Wales. See [1429].

  60. 60.

    The Davenport constant of a finite Abelian group G is the smallest integer m with the property that every sequence of m elements of G has a subsequence with vanishing sum.

  61. 61.

    Mikao Moriya (1906–1982), professor at Nakayama University.

  62. 62.

    This result has been also found by S. Chowla and Weil, but they did not publish it.

  63. 63.

    Morris Newman (1924–2007), professor of the University of California at Santa Barbara. See [1459].

  64. 64.

    Wolfgang Schwarz(1934–2013), professor in Freiburg and Frankfurt/M.

  65. 65.

    In case \(n=7\) the published proof is incomplete, but can be repaired. See the review of [4125] by Diaz y Diaz in Math. Reviews 84e:12005.

  66. 66.

    Axel Thue (1863–1922), professor in Oslo. See [366, 483].

  67. 67.

    Alfred Theodor Brauer (1894–1985), brother of Richard Brauer, professor at the University of North Carolina. See [1917, 3496].

  68. 68.

    Lubomir Tschakaloff (1886–1963), professor in Sofia.

  69. 69.

    Ludwig Holzer (1891–1968), professor in Vienna, Graz and Rostock.

  70. 70.

    Arthur Wieferich (1884–1954), teacher of mathematics.

  71. 71.

    Taro Morishima (1903–1989), professor in Tokyo.

  72. 72.

    John Barkley Rosser (1907–1989), professor at the University of Wisconsin in Madison.

  73. 73.

    George Pólya (1887–1985), professor at ETH in Zürich and at Stanford University. See [59, 4033].

  74. 74.

    Georges Fontené (1848–1923), teacher in Belfort, Douai, Rouen and at Collége Rollin in Paris. See [449].

  75. 75.

    Marie Ennemond Camille Jordan (1838–1922), professor in Paris. See [1852, 2494].

  76. 76.

    Walter Buckingham Carver (1879–1961), professor at the Cornell University.

  77. 77.

    Viggo Brun (1885–1978), professor in Oslo. See [3720].

  78. 78.

    Nils Johan Pipping (1890–1982), professor in Turku.

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Narkiewicz, W. (2018). First Years of the Century. 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_3

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