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Arithmetic in Some Quadratic Number Fields

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Quadratic Number Fields

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

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Abstract

In this chapter we present examples of norm-Euclidean quadratic number fields and apply the results to the Fermat equations with exponents 3, 4, and 5.

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Notes

  1. 1.

    See the beautiful article [72].

  2. 2.

    The gap is the one that we have pointed out in Chap. 1, namely the missing proof for the decomposition theorem for numbers of the form x 2 + 3y 2: If c 2 + 3d 2 = r 3, then there exist integers p and q with c = p(p 2 − 9q 2) and d = 3q(p 2 − q 2); see [10].

  3. 3.

    See https://mathoverflow.net/questions/39561.

  4. 4.

    This result is due to Theodore Motzkin (1908–1970) [98]. It can be proved quite easily and has played a big role in recent years; see Lemmermeyer [76].

  5. 5.

    The Dedekind–Hasse criterion was published by Helmut Hasse [57]. Emmy Noether later found this criterion among Dedekind’s papers when she edited his collected works [28]; see also [90, Anm. 1, S. 60].

  6. 6.

    Gauss formulated this conjecture for class numbers of binary quadratic forms with even middle coefficients.

References

  1. R. Ayoub, S. Chowla, On Euler’s polynomial. J. Numb. Theory 13, 443–445 (1981)

    Article  Google Scholar 

  2. C. Bergmann, Über Eulers Beweis des großen Fermatschen Satzes für den Exponenten 3. Math. Ann. 164, 159–175 (1966)

    Article  MathSciNet  Google Scholar 

  3. J.W.S. Cassels, Lectures on Elliptic Curves (Cambridge University Press, Cambridge, 1991)

    Book  Google Scholar 

  4. D. Cox, Why Eisenstein proved the Eisenstein criterion and why Schönemann discovered it first. Am. Math. Monthly 118, 3–21 (2011)

    Article  Google Scholar 

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

    Google Scholar 

  6. P.G.L. Dirichlet, Mémoire sur l’impossibilité de quelques équations indéterminées du cinquième degré. Acad. Sci. Royale France 1825; Werke I, 1–46

    Google Scholar 

  7. F.W. Dodd, Number theory in the integral domain \({\mathbb Z}[\frac 12 + \frac 12\sqrt {5}\,]\), Dissertation Univ. Northern Colorado, 1981; published as Number Theory in the Quadratic Field with Golden Section Unit (Polygonal Publishing House, Passaic, NJ, 1983)

    Google Scholar 

  8. R.B. Eggleton, C.B. Lacampagne, J.L. Selfridge, Euclidean quadratic fields. Am. Math. Monthly 99, 829–837 (1992)

    Article  MathSciNet  Google Scholar 

  9. F.G. Frobenius (unter Benutzung einer Mitteilung des Herrn Dr. R. Remak), Über quadratische Formen, die viele Primzahlen darstellen. Sitz. Kön. Preuß. Akad. Wiss. Berlin (1912), 966–980; Ges. Abh. III, 573–587

    Google Scholar 

  10. K. Halupczok, Euklidische Zahlkörper. Diplomarbeit (Hartung-Gorre Verlag, Konstanz, 1997)

    Google Scholar 

  11. G.H. Hardy, E.M. Wright, Einführung in die Zahlentheorie (R. Oldenbourg Verlag, München, 1958)

    MATH  Google Scholar 

  12. M. Harper, A proof that \({\mathbb Z}[\sqrt {14}\,]\) is Euclidean, Ph.D. thesis, McGill University, 2000

    Google Scholar 

  13. M. Harper, \({\mathbb Z}[\sqrt {14}\,]\) is Euclidean. Can. J. Math. 56, 55–70 (2004)

    Google Scholar 

  14. H. Hasse, Über eindeutige Zerlegung in Primelemente oder in Primhauptideale in Integritätsbereichen. J. Reine Angew. Math. 159, 3–12 (1928)

    MathSciNet  MATH  Google Scholar 

  15. E. Hecke, Lectures on the Theory of Algebraic Numbers (Springer, Berlin, 1981)

    Book  Google Scholar 

  16. D. Hilbert, Die Theorie der Algebraischen Zahlkörper, Jahresber. DMV 4, 175–546 (1897); Engl. Transl. I. Adamson, The Theory of Algebraic Number Fields (Springer, New York, 1998)

    Google Scholar 

  17. R.C. Laubenbacher, D. Pengelley, Eisenstein’s misunderstood geometric proof of the quadratic reciprocity theorem. College Math. J. 25, 29–34 (1994)

    Article  Google Scholar 

  18. F. Lemmermeyer, The Euclidean algorithm in algebraic number fields. Expositiones Math. 13, 385–416 (1995)

    MathSciNet  MATH  Google Scholar 

  19. F. Lemmermeyer, Reciprocity Laws (Springer, Berlin, 2000)

    Book  Google Scholar 

  20. F. Lemmermeyer, Composite values of irreducible polynomials. Elemente d. Math. 74, 36–37 (2019)

    Article  MathSciNet  Google Scholar 

  21. F. Lemmermeyer, P. Roquette (Hrsg.), Helmut Hasse und Emmy Noether – Die Korrespondenz 1925 – 1935 (Univ.-Verlag, Göttingen, 2006)

    MATH  Google Scholar 

  22. T. Motzkin, The Euclidean algorithm. Bull. Am. Math. Soc. 55, 1142–1146 (1949)

    Article  MathSciNet  Google Scholar 

  23. A. Oppenheim, Quadratic fields with and without Euclid’s algorithm. Math. Ann. 109, 349–352 (1934)

    Article  MathSciNet  Google Scholar 

  24. J. Plemelj, Die Unlösbarkeit von x 5 + y 5 + z 5 = 0 im Körper \(k\sqrt {5}\). Monatsh. Math. Phys. 23, 305–308 (1912)

    Google Scholar 

  25. G. Rabinovitch, Eindeutigkeit der Zerlegung in Primzahlfaktoren im quadratischen Zahlkörper. Proc. Int. Congr. Math. 1912, 418–421 (1912)

    Google Scholar 

  26. L. Rédei, Über die quadratischen Zahlkörper mit Primzerlegung. Acta Sci. Math. (Szeged) 21, 1–3 (1960)

    MathSciNet  MATH  Google Scholar 

  27. P. Ribenboim, Meine Zahlen, meine Freunde. Glanzlichter der Zahlentheorie (Springer, Berlin, 2009)

    Google Scholar 

  28. H. Siebeck, Die recurrenten Reihen, vom Standpuncte der Zahlentheorie aus betrachtet. J. Reine Angew. Math. 33, 71–77 (1846)

    MathSciNet  Google Scholar 

  29. J. Sommer, Vorlesungen über Zahlentheorie. Einführung in die Theorie der algebraischen Zahlkörper (Teubner, Leipzig, 1907)

    Google Scholar 

  30. G. Szekeres, On the number of divisors of x 2 + x + A, J. Number Theory 6, 434–442 (1974)

    Article  MathSciNet  Google Scholar 

  31. L. Tschakaloff, Unmöglichkeitsbeweis der Gleichung α 5 + β 5 = ηγ 5 im quadratischen Körper \(K(\sqrt {5}\,)\). Tôhoku Math. J. 27, 189–194 (1926)

    Google Scholar 

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Authors and Affiliations

  1. Jagstzell, Germany

    Franz Lemmermeyer

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  1. Franz Lemmermeyer
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Lemmermeyer, F. (2021). Arithmetic in Some Quadratic Number Fields. In: Quadratic Number Fields. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-030-78652-6_5

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