Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the number of real quadratic fields with class number divisible by 3


Authors: K. Chakraborty and M. Ram Murty
Journal: Proc. Amer. Math. Soc. 131 (2003), 41-44
MSC (2000): Primary 11R29; Secondary 11R11
Published electronically: May 15, 2002
MathSciNet review: 1929021
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We find a lower bound for the number of real quadratic fields whose class groups have an element of order $3$. More precisely, we establish that the number of real quadratic fields whose absolute discriminant is $\leq x$ and whose class group has an element of order $3$ is $\gg x^{\frac{5}{6}}$ improving the existing best known bound $\gg x^{\frac{1}{6}}$ of R. Murty.


References [Enhancements On Off] (What's this?)

  • 1. N. C. Ankeny and S. Chowla, On the divisibility of the class number of quadratic fields, Pacific J. Math. 5 (1955), 321–324. MR 0085301
  • 2. David A. Cardon and M. Ram Murty: Exponents of class groups of quadratic function fields over finite fields, Canadian Math. Bulletin, 44 (2001), 398-407.
  • 3. H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 33–62. MR 756082, 10.1007/BFb0099440
  • 4. H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405–420. MR 0491593
  • 5. P. Hartung, Proof of the existence of infinitely many imaginary quadratic fields whose class number is not divisible by 3, J. Number Theory 6 (1974), 276–278. MR 0352040
  • 6. Taira Honda, A few remarks on class numbers of imaginary quadratic number fields, Osaka J. Math. 12 (1975), 19–21. MR 0387240
  • 7. M. Ram Murty, Exponents of class groups of quadratic fields, Topics in number theory (University Park, PA, 1997) Math. Appl., vol. 467, Kluwer Acad. Publ., Dordrecht, 1999, pp. 229–239. MR 1691322
  • 8. K. Soundararajan, Divisibility of class numbers of imaginary quadratic fields, J. London Math. Soc. (2) 61 (2000), no. 3, 681–690. MR 1766097, 10.1112/S0024610700008887
  • 9. T. Nagell: Über die Klassenzahl imaginär quadratischer Zahlkorper: Abh. Math. Sem. Univ. Hamburg, 1 (1922), 140-150.
  • 10. P. J. Weinberger, Real quadratic fields with class numbers divisible by 𝑛, J. Number Theory 5 (1973), 237–241. MR 0335471
  • 11. Yoshihiko Yamamoto, On unramified Galois extensions of quadratic number fields, Osaka J. Math. 7 (1970), 57–76. MR 0266898

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11R29, 11R11

Retrieve articles in all journals with MSC (2000): 11R29, 11R11


Additional Information

K. Chakraborty
Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada K7L 3N6
Address at time of publication: Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, U. P., India
Email: kalyan@mast.queensu.ca, kalyan@mri.ernet.in

M. Ram Murty
Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada K7L 3N6
Email: murty@mast.queensu.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06603-0
Keywords: Class group, real quadratic fields
Received by editor(s): August 15, 2001
Published electronically: May 15, 2002
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 2002 American Mathematical Society