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Miscellaneous Results About the Class Number of Quadratic Fields

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Classical Theory of Algebraic Numbers

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Abstract

In this chapter, our purpose is to present a sample of results about the class number of quadratic fields. Due to their nature, several proofs have to be omitted. The reader is encouraged to study the original papers listed in the Bibliography.

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References

  1. Ankeny, N.C., Chowla, S., On the divisibility of the class number of quadratic fields, Pacific J. Math. 5 (1955), 321–324.

    Article  MathSciNet  MATH  Google Scholar 

  2. Baker, A., Imaginary quadratic fields with class number two, Ann. of Math. (2), 94 (1971), 139–157.

    Article  MathSciNet  MATH  Google Scholar 

  3. Baker, A., Transcendental Number Theory, Cambridge University Press, Cambridge, 1975.

    Book  MATH  Google Scholar 

  4. Deuring, M., Imaginär-quadratische Zahlkörper mit der Klassenzahl 1, Invent. Math. 5 (1968), 169–179.

    Article  MathSciNet  MATH  Google Scholar 

  5. Flath, D.E., Introduction to Number Theory, Wiley, New York, 1989.

    MATH  Google Scholar 

  6. Gauss, C.F., Disquisitiones Arithmeticae, originally published in 1801. Reprinted in many editions.

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  7. Goldfeld, D., Gauss’ class number problem for imaginary quadratic fields, Bull. Amer. Math. Soc. 134 (1985), 23–37.

    Article  MathSciNet  Google Scholar 

  8. Gross, B., Zagier, D., Heegner points and derivations of L-series, Invent. Math. 84 (1986), 225–320.

    Article  MathSciNet  MATH  Google Scholar 

  9. Gut, M., Die Zetafunktion, die Klassenzahl und die Kroneckersche Grenzformel eines beliebigen Kreiskörpers, Comment. Math. Helv.1 (1929), 160–226.

    Article  MathSciNet  MATH  Google Scholar 

  10. Gut, M., Kubische Klassenkörper über quadratischen imaginären Grundkörpern, Nieuw Arch. Wisk. (2), 23 (1951), 185–189.

    MathSciNet  MATH  Google Scholar 

  11. Gut, M., Erweiterungskörper von Primzahlgrad mit durch diese Primzahl teilbarer Klassenzahl, Enseign. Math. 19 (1973), 119–123.

    MathSciNet  MATH  Google Scholar 

  12. Hartung, P., Explicit construction of a class of infinitely many imaginary quadratic fields whose class number is divisible by 3, J. Number Theory 6 (1974), 279–281.

    Article  MathSciNet  MATH  Google Scholar 

  13. Heegner, K., Diophantische Analysis und Modulfunktionen, Math. Z. 56 (1952), 227–253.

    Article  MathSciNet  MATH  Google Scholar 

  14. Honda, T., On real quadratic fields whose class numbers are multiples of 3, J. Reine Angew. Math. 233 (1968), 101–102.

    MathSciNet  MATH  Google Scholar 

  15. Humbert, P., Sur les nombres de classes de certains corps quadratiques, Comment. Math. Helv. 12 (1940), 233–245.

    Article  MathSciNet  Google Scholar 

  16. Humbert, P., Note relative à l’article “Sur les nombres de classes de certains corps quadratiques” Comment. Math. Helv. 13 (1940), 67.

    Article  MathSciNet  Google Scholar 

  17. Kuroda, S., On the class-number of imaginary quadratic number fields, Proc. Japan Acad. Sci. 40 (1964), 365–367.

    Article  MathSciNet  MATH  Google Scholar 

  18. Nagell, T., Über die Klassenzahl imaginären quadratischer Zahlkörper, Abh. Math. Sem. Univ. Hamburg 1 (1922), 140–150.

    Article  MathSciNet  MATH  Google Scholar 

  19. Ribenboim, P., Gauss and the class number problem, Symp. Gaussiana1 (1991), 13–63

    Google Scholar 

  20. Stark, H.M., A complete characterization of the complex quadratic fields of class-number one, Michigan Math. J. 14 (1967), 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  21. Stark, H.M., On the “gap” in a theorem of Heegner, J. Number Theory 1 (1969), 16–27.

    Article  MathSciNet  MATH  Google Scholar 

  22. Yamamoto, Y., On unramified Galois extensions of quadratic number fields, Osaka J. Math. 7 (1970), 57–76.

    MathSciNet  MATH  Google Scholar 

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Ribenboim, P. (2001). Miscellaneous Results About the Class Number of Quadratic Fields. In: Classical Theory of Algebraic Numbers. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21690-4_28

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  • DOI: https://doi.org/10.1007/978-0-387-21690-4_28

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-2870-2

  • Online ISBN: 978-0-387-21690-4

  • eBook Packages: Springer Book Archive

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