Skip to main content
SpringerLink
Log in

Prehistory

  • Chapter
  • First Online:
Quadratic Number Fields

Part of the book series: Springer Undergraduate Mathematics Series ((SUMS))

  • 1695 Accesses

Abstract

This chapter presents bits of the history of unique factorization and relevant problems with quadratic surds from Euclid to Dedekind.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight 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.

    For learning more about the methods used in “Babylonian algebra” see [63].

  2. 2.

    For Euclid, the product of a number is the representation of a number as a product, not the result. When Euclid wants the result of a product, he uses a clumsy phrase such as “if two numbers multiplied make a number.”

  3. 3.

    Presentations of their work may be found in Vogel [126, 127] and in Chemla and Guo [19].

  4. 4.

    This included the Hellenistic world. Among the scientists believed to have studied in Alexandria are Archimedes from Syracuse in Sicily and Eratosthenes from Cyrene in North Africa. It is also conceivable that well-educated scribes from Mesopotamia preferred the boomtown Alexandria to the declining cities in Mesopotamia.

  5. 5.

    This is problem VI.17 in Heath [59]; some problems are enumerated in a different way in different editions.

  6. 6.

    Fermat means the sequence 2, 22 = 4, 42 = 16, 162 = 256, etc.

  7. 7.

    The notation a∣b stands for “a divides b”.

  8. 8.

    There were numerous less known mathematicians interested in Diophantine problems or the investigation of perfect and amicable numbers.

  9. 9.

    The norm of a Gaussian integer x + iy is (x + iy)(x − iy) = x2 + y2.

References

  1. G. Arendt, Éléments de la théorie des nombres complexes de la forme\(a+b\sqrt {-1}\). Programme Collège Royal Français, September 1863

    Google Scholar 

  2. K. Chemla, S. Guo, Les neuf chapitres. Le Classique mathématique de la Chine ancienne et ses commentaires (Dunod, Paris, 2004)

    Google Scholar 

  3. R. Dedekind, Gesammelte mathematische Werke (Friedrich Vieweg & Sohn, Braunschweig, 1932)

    Google Scholar 

  4. L. Euler, De numeris, qui sunt aggregata duorum quadratorum. Nova Comm. Acad. Sci. Petropol. 4(1752/3), 1758, 3–40; Opera Omnia I - 2, 295–327

    Google Scholar 

  5. L. Euler, Vollständige Anleitung zur Algebra (Birkhäuser, Basel, 1770); Leipzig (1883)

    Google Scholar 

  6. L. Euler, Opera Postuma. Fragmenta arithmetica ex adversariis mathematicis deprompta, vol. 1, (1862), pp. 231–232; S. 157

    Google Scholar 

  7. C.F. Gauß, Disquisitiones Arithmeticae, 1801; deutsche Übersetzung Maser 1889; Neuauflage (K. Reich, Hrsg.), Georg Olms Verlag 2015

    Google Scholar 

  8. C.F. Gauß, Theorie der biquadratischen Reste. Zweite Abhandlung (Göttingen, 1832); deutsche Übersetzung Maser 1889

    Google Scholar 

  9. T.L. Heath, Diophantus of Alexandria. A Study in the History of Greek Algebra (Cambridge University Press, Cambridge, 1910)

    Google Scholar 

  10. J. Høyrup, Algebra in Cuneiform (Max-Planck-Gesellschaft zur Förderung der Wissenschaften, Berlin, 2017)

    MATH  Google Scholar 

  11. H.W.E. Jung, Einführung in die Theorie der quadratischen Zahlkörper (Jänicke, Leipzig, 1936)

    MATH  Google Scholar 

  12. E.E. Kummer, Zur Theorie der complexen Zahlen. J. Reine Angew. Math. 35, 319–326 (1847)

    MathSciNet  Google Scholar 

  13. F. Lemmermeyer, Zur Zahlentheorie der Griechen. I: Euklids Fundamentalsatz der Arithmetik. Math. Semesterber. 55, 181–195 (2008)

    MathSciNet  MATH  Google Scholar 

  14. F. Lemmermeyer, Jacobi and Kummer’s ideal numbers. Abh. Math. Sem. Hamburg 79, 165–187 (2009)

    Article  MathSciNet  Google Scholar 

  15. F. Lemmermeyer, 4000 Jahre Zahlentheorie (Springer-Verlag, to appear)

    Google Scholar 

  16. F. Lemmermeyer, M. Mattmüller (Hrsg.), Leonhardi Euler Opera Omnia (IV)4. Correspondence of Leonhard Euler with Christian Goldbach (Birkhäuser, Basel, 2015)

    Google Scholar 

  17. D. Pengelley, F. Richman, Did Euclid need the Euclidean algorithm to prove unique factorization?. Am. Math. Monthly 113, 196–205 (2006)

    Article  MathSciNet  Google Scholar 

  18. E. Trost, Eine Bemerkung zur diophantischen Analysis. Elem. Math. 26, 60–61 (1971)

    MathSciNet  MATH  Google Scholar 

  19. E. Trost, Solution of Problem E 2332. Am. Math. Monthly 87, p. 77 (1972)

    Google Scholar 

  20. K. Vogel, Vorgriechische Mathematik II. Die Mathematik der Babylonier (Schroedel, Hannover, 1959)

    Google Scholar 

  21. K. Vogel (Hrsg.), Neun Bücher arithmetischer Technik (Ostwalds Klassiker der exakten Naturwissenschaften, Braunschweig, 1968)

    Google Scholar 

  22. A. Weil, Number Theory: An Approach Through History from Hammurapi to Legendre (Birkhäuser, New York, 1984)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Jagstzell, Germany

    Franz Lemmermeyer

Authors
  1. Franz Lemmermeyer
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Lemmermeyer, F. (2021). Prehistory. In: Quadratic Number Fields. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-030-78652-6_1

Download citation

Publish with us

Policies and ethics

Search

Navigation