Skip to main content
SpringerLink
Log in

The Twenties

  • Chapter
  • First Online:
The Story of Algebraic Numbers in the First Half of the 20th Century

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

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 79.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Max Noether (1844–1921), professor in Heidelberg and Erlangen, father of Emmy Noether. See [452, 2675].

  2. 2.

    Rings having this property are now called Noetherian rings .

  3. 3.

    Francis Sowerby Macaulay (1862–1937), teacher at St Paul’s School in London. See [182].

  4. 4.

    Werner Schmeidler (1890–1969), professor at the Technische Hochschule Breslau and Technische Universität Berlin.

  5. 5.

    Irvin Sol Cohen (1917–1955), professor at the Massachusetts Institute of Technology.

  6. 6.

    Heinz Prüfer (1896–1934), professor in Münster. See [273].

  7. 7.

    John von Neumann (1903–1957), professor at the Institute in Princeton. See [4101]

  8. 8.

    Harold Davenport (1907–1969), professor in Bangor, at the University College London and in Cambridge. See [2939, 2940, 3489, 3491].

  9. 9.

    It follows from the remarks made by Cohn in [745] that Davenport and Heilbronn obtained these inequalities at least 15 years earlier.

  10. 10.

    There are two Theorems 4 in Herbrand’s paper.

  11. 11.

    In the introduction Ore stated that his method works in all cases, but this had not been substantiated by the content of the paper.

  12. 12.

    Ernst Jacobsthal (1882–1965), professor in Berlin and Trondheim. See [3726].

  13. 13.

    This argument occurs on p. 207 of [3768] in the case of the equation \(u-v=c\).

  14. 14.

    William Judson LeVeque (1923–2007), professor at the University of Michigan. See [2790].

  15. 15.

    Klaus Friedrich Roth, (1925–2015), professor at the University College and Imperial College in London.

  16. 16.

    Julia Robinson (1919–1985), professor at Berkeley. See [3421].

  17. 17.

    Actually for the equivalent equation \(u+v=-1\).

  18. 18.

    Veikko Ennola (1932–2013), professor in Turku.

  19. 19.

    Vladimir Genad’evič Sprindžuk (1936–1987), professor in Minsk.

  20. 20.

    Béla Brindza (1958–2003), professor in Debrecen.

  21. 21.

    Leo Felix Pollaczek (1892–1981) worked in a telephone company in Germany and later in France.

  22. 22.

    Hendrik Douwe Kloosterman (1900–1968), professor in Leiden. See [3881].

  23. 23.

    Erich Lamprecht (1926–2003), professor in Saarbrücken.

  24. 24.

    Adolf Krazer (1858–1926), professor at the Technische Hochschule in Karlsruhe.

  25. 25.

    Marie Georges Humbert (1859–1921), professor at École Polytechnique in Paris. See [2493].

  26. 26.

    Harvey Cohn (1923–2014), professor at the University of Arizona, CUNY and Stanford.

  27. 27.

    Hans Maass (1911–1992), professor in Heidelberg. See [524].

  28. 28.

    Godfrey Harold Hardy (1877–1947), professor in Oxford and Cambridge. See [4066].

  29. 29.

    Srinivasa Aiyangar Ramanujan (1887–1920), Fellow of Trinity College, Cambridge. See [80, 1625].

  30. 30.

    Erich Kamke (1890–1961), professor in Tübingen. See [4284].

  31. 31.

    Loo-Keng Hua (1910–1985), professor in Beijing. See [4295].

  32. 32.

    Anatoliĭ Alekseevič Karatsuba (1937–2008), professor in Moscow. See [653].

  33. 33.

    Sergeĭ Borisovič Stečkin (1920–1995), professor in Sverdlovsk and Moscow. See [331].

  34. 34.

    Askold Ivanovič Vinogradov (1929–2005), professor in the Steklov Institute.

  35. 35.

    Chen Jing Run (1933–1996), professor in Beijing.

  36. 36.

    Heini Halberstam (1926–2014), professor in Dublin, Nottingham, and at the University of Illinois at Urbana-Champaign.

  37. 37.

    Gabor Szegő (1895–1985), professor in Berlin, Königsberg, at the Washington University in St. Louis and at Stanford. See [149].

  38. 38.

    Zyoiti Suetuna (1898–1970), professor in Kyushu University and in Tokyo University.

  39. 39.

    Heinrich Behnke (1898–1979), professor in Münster. See [1502].

  40. 40.

    Yoshihiko Yamamoto (1941–2004), professor in Osaka.

  41. 41.

    Fritz Grunewald (1949–2010), professor in Bonn and Düsseldorf. See [3723].

  42. 42.

    It seems that it has been formulated in print on p. 46 of [1655].

  43. 43.

    Alekseĭ Ivanovič Kostrikin (1929–2000), professor in Moscow.

  44. 44.

    Boris Borisovič Venkov (1934–2011), son of B.A. Venkov, professor in St. Petersburg and Aachen.

  45. 45.

    Artin considered also the case when the base field is an arbitrary finite field, but did not publish his results, and his notes on this subject appeared much later, in 2000 [131].

  46. 46.

    Paul Sengenhorst (1894–1968). See [272].

  47. 47.

    Let us recall that a group is metabelian if its commutator group is Abelian.

  48. 48.

    Ilya Piatetski-Shapiro (1929–2009), professor in Moscow, Tel Aviv and at Yale. See [730].

  49. 49.

    Joseph Shalika (1941–2010), professor at John Hopkins University.

  50. 50.

    Felix Pollaczek (1892–1981) worked in Berlin and Paris as engineer.

  51. 51.

    Richard Dagobert Brauer (1901–1977), brother of Alfred Brauer, professor in Toronto, Ann Arbor and at Harvard. See [1168, 1508, 3495].

  52. 52.

    Bernard Morris Dwork (1923–1998), professor at the Johns Hopkins University and Princeton University. See [2123].

  53. 53.

    Otto Schreier (1901–1929), professor in Rostock. See [2813].

  54. 54.

    I owe to the important paper of Mr. Tschebotareff one of the fundamental ideas of the proof, the use of extensions of cyclotomic fields.

  55. 55.

    Wilhelm Magnus (1907–1990), professor in Göttingen, Courant Institute and Polytechnic Institute of New York.

  56. 56.

    Ernst Witt (1911–1991), professor in Hamburg. See [2139].

  57. 57.

    Zenon Ivanovič Borevič (1922–1995), professor in Leningrad. See [3073].

  58. 58.

    Tadao Tannaka (1908–1986), professor at the Tôhoku University. See [14].

  59. 59.

    Tonny Albert Springer (1926–2011), professor at the Technical University Utrecht. See [4124].

  60. 60.

    Wilhelm Grunwald (1909–1989) worked in Göttingen.

  61. 61.

    Manohar Lal Madan (1935–2011), professor at Ohio State University.

  62. 62.

    Luther Elic Claborn (1935–1967), professor at the University of Illinois in Urbana.

  63. 63.

    Frank Gerth III (1945–2006), professor at the University of Texas in Austin.

  64. 64.

    Kurt Werner Reidemeister (1893–1971), professor in Königsberg, Marburg and Göttingen. See [141].

  65. 65.

    James Burton Ax (1937–2006), professor at Cornell University and in Stony Brook.

  66. 66.

    Walter Feit (1920–2004), professor at Yale. See [3719].

  67. 67.

    Gennadiĭ Vladimirovič Belyĭ (1951–2001) worked in Vladimir University. See [377].

  68. 68.

    Michael Fekete (1886–1957), professor at the Hebrew University in Jerusalem. See [3492].

  69. 69.

    Raphael Mitchell Robinson (1911–1995), professor at Berkeley. See [1761].

  70. 70.

    Theodore Samuel Motzkin (1908–1970), professor at the Hebrew University in Jerusalem and the University of California in Los Angeles. See [9].

  71. 71.

    George Greaves (1941–2008), professor in Cardiff. See [1728].

  72. 72.

    Robert Winston Keith Odoni (1947–2002), professor in Exeter and Glasgow. See [742].

  73. 73.

    Jean Favard (1902–1965), professor in Grenoble and at the l’École Polytechnique in Paris.

  74. 74.

    Erich Stiemke (1892–1915), killed in World War I.

  75. 75.

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

  76. 76.

    Dörge wrote in [991] that this condition has been also known to Weil.

  77. 77.

    Friedrich Wilhelm Neuhaus (1899–1983), professor in Köln.

  78. 78.

    Donald J. Lewis (1926–2015), professor at the Notre Dame University and the University of Michigan.

  79. 79.

    Alfred Jacobus van der Poorten (1942–2010), professor at UNSW in Sydney and at the Macquarie University in North Ryde.

  80. 80.

    Robert Fricke (1861–1930), professor in Braunschweig.

  81. 81.

    He pointed out that it is based on Weber’s book [1302].

Author information

Authors and Affiliations

  1. University of Wrocław, Wrocław, Poland

    Władysław Narkiewicz

Authors
  1. Władysław Narkiewicz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Władysław Narkiewicz .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Narkiewicz, W. (2018). The Twenties. 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_4

Download citation

Publish with us

Policies and ethics

Search

Navigation