Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T17:00:21.981Z Has data issue: false hasContentIssue false
  • English
  • Français

A Lower Bound on the Number of Cyclic Function Fields With Class Number Divisible by n

Published online by Cambridge University Press:  20 November 2018

Allison M. Pacelli
Allison M. Pacelli*
Affiliation:
Department of Mathematics, Williams College, Williamstown, MA 01267 e-mail: Allison.Pacelli@williams.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we find a lower bound on the number of cyclic function fields of prime degree $l$ whose class numbers are divisible by a given integer $n$. This generalizes a previous result of D. Cardon and R. Murty which gives a lower bound on the number of quadratic function fields with class numbers divisible by $n$.

Keywords

11R2911R58
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 3 , 01 September 2006 , pp. 448 - 463
Copyright
Copyright © Canadian Mathematical Society 2006

References

[1] Byeon, D. and Koh, E., Real quadratic fields with class number divisible by 3. Manuscripta Math. 111(2003), no. 2, 261–263.Google Scholar
[2] Cardon, D. and Ram Murty, R., Exponents of class groups of quadratic function fields over finite fields. Canad. Math. Bull. 44(2001), no. 4, 398–407.Google Scholar
[3] Chakraborty, K. and Ram Murty, M., On the number of real quadratic fields with class number divisible by 3. Proc. Amer. Math. Soc. 131(2003), no. 1, 41–44.Google Scholar
[4] Cohen, H. and Lenstra, H. W. Jr., Heuristics on class groups of number fields. In: Number Theory, Lecture Notes in Math. 1068, Springer, Berlin, 1984, pp. 33–62.Google Scholar
[5] Cohen, H. and Martinet, J., Class groups of number fields: numerical heuristics. Math. Comp. 48(1987), no. 177, 123–137.Google Scholar
[6] Friedman, E. and Washington, L. C., On the distribution of divisor class groups of curves over a finite field. In: Théorie des nombres, de Gruyter, Berlin, 1989, pp. 227–239.Google Scholar
[7] Friesen, C., Class number divisibility in real quadratic function fields. Canad. Math. Bull. 35(1992), no. 3, 361–370.Google Scholar
[8] Gauss, C. F., Disquisitiones Arithmeticae. Leipzig, 1801.Google Scholar
[9] Kummer, E., Beweis des Fermat’schen Satzes der Unmöglichkeit von xλ + yλ = zλ für eine unendliche [sic] Anzahl Primzahlen λ . Monatsber. Akad. Wiss. Berlin, 1847, 132–141, 305-319.Google Scholar
[10] Luca, F., A note on the divisibility of class numbers of real quadratic fields. C. R. Math. Acad. Sci. Soc. R. Can. 25(2003), no. 3, 71–75.Google Scholar
[11] Murty, M. R., Exponents of class groups of quadratic fields. In: Topics in Number Theory, Math. Appl. 467, Kluwer Academic. Publishers, Dordrecht, 1999, pp. 229–239.Google Scholar
[12] Nagell, T., Uber die Klassenzahl imaginar quadratischer Zahlkorper. Abh. Math. Sem. Univ. Hamburg 1(1922), 140–150.Google Scholar
[13] Rosen, M., Average value of class numbers in cyclic extensions of the rational function field. In: Number Theory, CMS Conf. Proc. 15, American Mathematical Society, Providence, RI, 1995, pp. 307–323.Google Scholar
[14] Soundararajan, K., Divisibility of class numbers of imaginary quadratic fields. J. London Math. Soc. 61(2000), no. 3, 681–690.Google Scholar
[15] Yamamoto, Y., On unramified Galois extensions of quadratic number fields. Osaka J. Math. 7(1970), 57–76.Google Scholar
[16] Yu, G., A note on the divisibility of class numbers of real quadratic fields. J. Number Theory 97(2002), no. 1, 35–44.Google Scholar