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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ankeny, N.C., Chowla, S., On the divisibility of the class number of quadratic fields, Pacific J. Math. 5 (1955), 321–324.
Baker, A., Imaginary quadratic fields with class number two, Ann. of Math. (2), 94 (1971), 139–157.
Baker, A., Transcendental Number Theory, Cambridge University Press, Cambridge, 1975.
Deuring, M., Imaginär-quadratische Zahlkörper mit der Klassenzahl 1, Invent. Math. 5 (1968), 169–179.
Flath, D.E., Introduction to Number Theory, Wiley, New York, 1989.
Gauss, C.F., Disquisitiones Arithmeticae, originally published in 1801. Reprinted in many editions.
Goldfeld, D., Gauss’ class number problem for imaginary quadratic fields, Bull. Amer. Math. Soc. 134 (1985), 23–37.
Gross, B., Zagier, D., Heegner points and derivations of L-series, Invent. Math. 84 (1986), 225–320.
Gut, M., Die Zetafunktion, die Klassenzahl und die Kroneckersche Grenzformel eines beliebigen Kreiskörpers, Comment. Math. Helv.1 (1929), 160–226.
Gut, M., Kubische Klassenkörper über quadratischen imaginären Grundkörpern, Nieuw Arch. Wisk. (2), 23 (1951), 185–189.
Gut, M., Erweiterungskörper von Primzahlgrad mit durch diese Primzahl teilbarer Klassenzahl, Enseign. Math. 19 (1973), 119–123.
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.
Heegner, K., Diophantische Analysis und Modulfunktionen, Math. Z. 56 (1952), 227–253.
Honda, T., On real quadratic fields whose class numbers are multiples of 3, J. Reine Angew. Math. 233 (1968), 101–102.
Humbert, P., Sur les nombres de classes de certains corps quadratiques, Comment. Math. Helv. 12 (1940), 233–245.
Humbert, P., Note relative à l’article “Sur les nombres de classes de certains corps quadratiques” Comment. Math. Helv. 13 (1940), 67.
Kuroda, S., On the class-number of imaginary quadratic number fields, Proc. Japan Acad. Sci. 40 (1964), 365–367.
Nagell, T., Über die Klassenzahl imaginären quadratischer Zahlkörper, Abh. Math. Sem. Univ. Hamburg 1 (1922), 140–150.
Ribenboim, P., Gauss and the class number problem, Symp. Gaussiana1 (1991), 13–63
Stark, H.M., A complete characterization of the complex quadratic fields of class-number one, Michigan Math. J. 14 (1967), 1–27.
Stark, H.M., On the “gap” in a theorem of Heegner, J. Number Theory 1 (1969), 16–27.
Yamamoto, Y., On unramified Galois extensions of quadratic number fields, Osaka J. Math. 7 (1970), 57–76.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media New York
About this chapter
Cite this chapter
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
Download citation
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