Skip to main content

Gauß und das Klassenzahlproblem

  • Chapter
Meine Zahlen, meine Freunde

Part of the book series: Springer-Lehrbuch ((SLB))

  • 1772 Accesses

Zusammenfassung

Die Theorie der binären quadratischen Formen gehört zu den großen Errungenschaften von Gauss in der Zahlentheorie. Einige der von Gauss formulierten Vermutungen sind noch heute Gegenstand umfangreicher Forschung. Dieser Text enthält auch eine kurze Beschreibung der wichtigsten Ergebnisse der jüngeren Vergangenheit bezüglich der Vermutungen von Gauss über die Klassenzahl.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 29.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 39.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literaturverzeichnis

  • 1801 C. F. Gauss. Disquisitiones Arithmeticae. G. Fleischer, Leipzig.

    Google Scholar 

  • 1870 C. F. Gauss. Werke. Königl. Ges. d. Wiss., Göttingen.

    Google Scholar 

  • 1892 P. Bachmann. Zahlentheorie, Band I und II. B. G. Teubner, Leipzig.

    Google Scholar 

  • 1907 J. Sommer. Vorlesungen über Zahlentheorie. B. G. Teubner, Leipzig.

    Google Scholar 

  • 1913 E. Landau. Über die Klassenzahl imaginär-quadratischer Zahlkörper. Göttinger Nachr., 285–295.

    Google Scholar 

  • 1929 T. Nagell. Über die Klassenzahl imaginär-quadratischer Zahlkörper. Abh. Math. Sem. Univ. Hamburg, 1:140–150.

    Google Scholar 

  • 1933 M. Deuring. Imaginäre quadratische Zahlkörper mit der Klassenzahl 1. Math. Z., 37:405–415.

    Google Scholar 

  • 1934 S. Chowla. An extension of Heilbronn’s class-number theorem. Quart. J. Math. Oxford, 5:304–307.

    Google Scholar 

  • 1934 H. Heilbronn. On the class number of imaginary quadratic fields. Quart. J. Math. Oxford, 5(2):150–160.

    Google Scholar 

  • 1934 H. Heilbronn und E. H. Linfoot. On the imaginary quadratic corpora of class number one. Quart. J. Math. Oxford, 5(2):293–301.

    Google Scholar 

  • 1936 C. L. Siegel. Über die Classenzahl quadratischer Zahlkörper. Acta Arith., 1:83–86. Nachdruck in Gesammelte Abhandlungen, Vol. I, 406–409. Springer-Verlag, Berlin, 1966.

    Google Scholar 

  • 1939 P. Humbert. Sur les nombres de classes de certains corps quadratiques. Comm. Math. Helvetici, 12:233–245 und 13:67 (1940).

    Google Scholar 

  • 1952 K. Heegner. Diophantische Analysis und Modulfunktionen. Math. Z., 56:227–253.

    Google Scholar 

  • 1954 S. Chowla und W. E. Briggs. On discriminants of binary quadratic forms with a single class in each genus. Can. J. Math., 6: 463–470.

    Google Scholar 

  • 1955 N. C. Ankeny und S. Chowla. On the divisibility of the class number of quadratic fields. Pacific J. Math., 5:321–324.

    Google Scholar 

  • 1961 G. B. Mathews. Theory of Numbers. Nachdruck von Chelsea Publ. Co., Bronx, NY.

    Google Scholar 

  • 1962 H. Cohn. Advanced Number Theory. Dover, New York.

    Google Scholar 

  • 1963 E. Grosswald. Negative discriminants of binary quadratic forms with one class in each genus. Acta Arith., 8:295–306.

    Google Scholar 

  • 1966 A. Baker. Linear forms in the logarithms of algebraic numbers. Mathematika, 13:204–216.

    Google Scholar 

  • 1966 Z. I. Borevich und I. R. Shafarevich. Number Theory. Academic Press, New York.

    Google Scholar 

  • 1966 J. Steinig. On Euler’s idoneal numbers. Elem. of Math., 21: 73–96.

    Google Scholar 

  • 1967 H. M. Stark. A complete determination of the complex quadratic fields of class number one. Michigan Math. J., 14:1–27.

    Google Scholar 

  • 1968 M. Deuring. Imaginäre-quadratische Zahlkörper mit der Klassenzahl Eins. Invent. Math., 5:169–179.

    Google Scholar 

  • 1968 T. Honda. On real quadratic fields whose class numbers are multiples of 3. J. reine u. angew. Math., 233:101–102.

    Google Scholar 

  • 1968 P. G. Lejeune-Dirichlet. Vorlesungen über Zahlentheorie (mit Zusätzen versehen von R. Dedekind). Chelsea Publ. Co., New York. Nachdruck. Erste Ausgabe 1863.

    Google Scholar 

  • 1968 C. L. Siegel. Zum Beweise des Starkschen Satz. Invent. Math., 5:180–191.

    Google Scholar 

  • 1969 D. Shanks. Class number, a theory of factorization, and genera. In 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), 415–440, Providence, R.I. Amer. Math. Soc.

    Google Scholar 

  • 1969 H. M. Stark. On the “gap” in a theorem of Heegner. J. Nb. Th., 1:16–27.

    Google Scholar 

  • 1970 B. A. Venkov. Elementary Number Theory. Wolters-Noordhoff Publishing, Gröningen. Übersetzt aus dem Russischen und überarbeitet von H. Alderson.

    Google Scholar 

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

    Google Scholar 

  • 1971 A. Baker. Imaginary quadratic fields with class number 2. Ann. of Math. (2), 94:139–152.

    Google Scholar 

  • 1971 H. M. Stark. A transcendence theorem for class number problems. Ann. Math. (2), 94:153–173.

    Google Scholar 

  • 1972 D. W. Boyd und H. Kisilevsky. On the exponent of the ideal class groups of complex quadratic fields. Proc. Amer. Math. Soc., 31:433–436.

    Google Scholar 

  • 1973 P. J. Weinberger. Exponents of the class groups of complex quadratic fields. Acta Arith., 22:117–124.

    Google Scholar 

  • 1973 P. J. Weinberger. Real quadratic fields with class numbers divisible by n. J. Nb. Th., 5:237–241.

    Google Scholar 

  • 1975 A. Baker. Transcendental Number Theory. Cambridge Univ. Press, Cambridge.

    Google Scholar 

  • 1976 D. A. Buell. Class groups of quadratic fields. Math. of Comp., 30:610–623.

    Google Scholar 

  • 1976 S. Chowla und J. B. Friedlander. Some remarks on L-functions and class numbers. Acta Arith., 28:414–417.

    Google Scholar 

  • 1976 D. Shanks. A survey of quadratic, cubic and quartic algebraic number fields (from a computational point of view). In Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory, and Computing (Louisiana State Univ., Baton Rouge, LA), 15–40. Utilitas Math., Winnipeg, Manitoba.

    Google Scholar 

  • 1977 M. Craig. A construction for irregular discriminants. Osaka J. Math., 14:365–402.

    Google Scholar 

  • 1977 H. M. Edwards. Fermat’s Last Theorem: A Genetic Introduction to Algebraic Number Theory. Springer-Verlag, New York.

    Google Scholar 

  • 1977 D. M. Goldfeld. The conjectures of Birch and Swinnerton-Dyer and the class numbers of quadratic fields. Astérisque 41–42, 219–227.

    Google Scholar 

  • 1980 H. Davenport. Multiplicative Number Theory. Springer-Verlag, New York, 2. Ausgabe.

    Google Scholar 

  • 1980 A. Schinzel. On the relation between two conjectures on polynomials. Acta Arith., 38:285–322.

    Google Scholar 

  • 1981 W. Kaufmann-Bühler. Gauss: A Biographical Study. Springer-Verlag, Berlin-Heidelberg-New York.

    Google Scholar 

  • 1981 H. Wada. A table of ideal class numbers of real quadratic fields. Sophia Kokyoroku in Mathematics. Number 10.

    Google Scholar 

  • 1981 D. B. Zagier. Zetafunktionen und quadratische Körper. Springer-Verlag, Berlin.

    Google Scholar 

  • 1982 L. K. Hua. Introduction to Number Theory. Springer-Verlag, Berlin.

    Google Scholar 

  • 1983 B. Gross und D. B. Zagier. Points de Heegner et dérivées de fonctions L. C. R. Acad. Sci. Paris, 297:85–87.

    Google Scholar 

  • 1983 J. Oesterlé. Nombres de classes des corps quadratiques imaginaires. Séminaire Bourbaki, exp. 631.

    Google Scholar 

  • 1983 R. J. Schoof. Class groups of complex quadratic fields. Math. of Comp., 41:295–302.

    Google Scholar 

  • 1984 H. Cohen und H. W. Lenstra, Jr. Heuristics on class groups of number fields. In Number Theory, Noordwijkerhout 1983, Lect. Notes in Math., 1068, 33–62. Springer-Verlag, Berlin.

    Google Scholar 

  • 1984 G. Frei. Les nombres convenables de Leonhard Euler. Sém. Th. des Nombres), Besan¸con, (1983–84). 58 Seiten.

    Google Scholar 

  • 1984 J. J. Gray. A commentary on Gauss’s mathematical diary, 1796–1814, with an English translation. Expo. Math., 2:97–130.

    Google Scholar 

  • 1984 A. Weil. Number theory, an Approach through History, from Hammurapi to Legendre. Birkhäuser, Boston.

    Google Scholar 

  • 1984 D. B. Zagier. L-series of elliptic curves, the Birch-Swinnerton-Dyer conjecture, and the class number problem of Gauss. Notices Amer. Math. Soc., 31(7):739–743.

    Google Scholar 

  • 1985 G. Frei. Leonhard Euler’s convenient numbers. Math. Intelligencer, 7(3):55–58, 64.

    Google Scholar 

  • 1985 D. M. Goldfeld. Gauss’s class number problem for imaginary quadratic fields. Bull. Amer. Math. Soc., 13:23–37.

    Google Scholar 

  • 1985 J. P. Serre. = b 2 − 4ac. Mathematical Medley, 13(1):1–10. Siehe auch den Anhang in Flath (1989).

    Google Scholar 

  • 1986 B. Gross und D. B. Zagier. Heegner points and derivatives of L-series. Invent. Math., 84:225–320.

    Google Scholar 

  • 1986 G. Lachaud. Sur les corps quadratiques réels principaux. In Séminaire de Théorie des Nombres, Paris 1984–85. Progress in Math. #63, 165–175. Birkhäuser Boston, Boston, MA.

    Google Scholar 

  • 1986 R. A. Mollin. On class numbers of quadratic extensions of algebraic number fields. Proc. Japan Acad., Ser. A, 62:33–36.

    Google Scholar 

  • 1986 R. Sasaki. A characterization of certain real quadratic fields. Proc. Japan Acad. Ser. A Math. Sci., 62:97–100.

    Google Scholar 

  • 1987 J. M. Borwein und P. B. Borwein. Pi and the AGM. John Wiley & Sons, New York.

    Google Scholar 

  • 1987 D. A. Buell. Class groups of quadratic fields, II. Math. of Comp., 48:85–93.

    Google Scholar 

  • 1987 G. Lachaud. On real quadratic fields. Bull. Amer. Math. Soc., 17:307–311.

    Google Scholar 

  • 1987 J. Quer. Corps quadratiques de 3-rang 6 et courbes elliptiques de rang 12. C. R. Acad. Sci. Paris, 305(6):215–218.

    Google Scholar 

  • 1988 P. Llorente und J. Quer. On the 3-Sylow subgroup of the class group of quadratic fields. Math. of Comp., 50:321–333.

    Google Scholar 

  • 1988 R. A. Mollin und H. C. Williams. A conjecture of S. Chowla via the generalized Riemann hypothesis. Proc. Amer. Math. Soc., 102:794–796.

    Google Scholar 

  • 1988 J. Oesterlé. Le problème de Gauss sur le nombres de classes. L’Enseign. Math., 2e série, 34:43–67.

    Google Scholar 

  • 1988 M. Saito und H. Wada. A table of ideal class groups of imaginary quadratic fields. Sophia Kokyoroku in Mathematics. Number 28.

    Google Scholar 

  • 1988 M. Saito und H. Wada. Tables of ideal class groups of real quadratic fields. Proc. Japan Acad., Ser. A, 64:347–349.

    Google Scholar 

  • 1989 D. A. Buell. Binary Quadratic Forms. Springer-Verlag, New York.

    Google Scholar 

  • 1989 D. A. Cox. Primes of the Form x 2 + ny 2. Wiley-Interscience, New York.

    Google Scholar 

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

    Google Scholar 

  • 1989 R. A. Mollin und H. C. Williams. Real quadratic fields of class number one and continued fraction period less than six. C. R. Math. Reports Acad. Sci. Canada, 11:51–56.

    Google Scholar 

  • 1989 D. Shanks. On Gauss and composition, I and II. In Proc. Conf. Canadian Nb. Th. Assoc., Banff, herausgegeben von R. A. Mollin, 163–204. Kluwer Acad. Publ., Dordrecht.

    Google Scholar 

  • 1990 R. A. Mollin und H. C. Williams. Class number problems for real quadratic fields. In Number Theory and Cryptography (Sydney, 1989), 177–195. Cambridge Univ. Press, Cambridge. In London Math. Soc. Lecture Notes Ser., 154.

    Google Scholar 

  • 1992 R. A. Mollin und H. C. Williams. Computation of the class number of a real quadratic field. Utilitas Math., 41:259–308.

    Google Scholar 

  • 1993 H. Cohen. A Course in Computational Algebraic Number Theory. Springer-Verlag, Berlin.

    Google Scholar 

  • 1996 R. A. Mollin. Quadratics. CRC Press, Boca Raton, FL.

    Google Scholar 

  • 1998 M. J. Jacobson, Jr. Experimental results on class groups of real quadratic fields (extended abstract). In Algorithmic Number Theory (Portland, OR, 1998), 463–474. Springer-Verlag, Berlin.

    Google Scholar 

Download references

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

(2009). Gauß und das Klassenzahlproblem. In: Meine Zahlen, meine Freunde. Springer-Lehrbuch. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87957-2_6

Download citation

Publish with us

Policies and ethics