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 exponent of the ideal class groups of complex quadratic fields.


Authors: David W. Boyd and H. Kisilevsky
Journal: Proc. Amer. Math. Soc. 31 (1972), 433-436
DOI: https://doi.org/10.1090/S0002-9939-1972-0289454-4
MathSciNet review: 0289454
Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: Let $ m(d)$ denote the exponent of the ideal class group of the complex quadratic field $ Q(\surd d)$, where $ d < 0$ is a fundamental discriminant. It is shown that there are only finitely many $ d$ for which $ m(d) = 3$. Assuming the extended Riemann Hypothesis, it is shown that $ m(d) \to \infty {\text{ as }}\vert d\vert \to \infty $.


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

  • [1] N. C. Ankeny, The least quadratic non residue, Ann. of Math. (2) 55 (1952), 65-72. MR 13, 538. MR 0045159 (13:538c)
  • [2] Z. I. Borevič and I. R. Šafarevič, Number theory, ``Nauka", Moscow, 1964; English transl., Pure and Appl. Math., vol. 20, Academic Press, New York, 1966. MR 30 #1080; MR 33 #4001. MR 0195803 (33:4001)
  • [3] S. Chowla, An extension of Heilbronn's class number theorem, Quart. J. Math. 5 (1934), 304-307.
  • [4] J. E. Littlewood, On the class number of the corpus $ P(\surd - k)$, Proc. London Math. Soc. 27 (1927), 358-372.
  • [5] H. Montgomery, Topics in multiplicative number theory, Thesis, Cambridge University, Cambridge, 1971. MR 0337847 (49:2616)
  • [6] D. Shanks, New types of quadratic fields having invariants divisible by three, J. Number Theory (to appear). MR 0313220 (47:1775)
  • [7] A. I. Vinogradov and Ju. V. Linnik, Hyperelliptic curves and the least prime quadratic residue, Dokl. Akad. Nauk SSSR 168 (1966), 259-261 = Soviet Math. Dokl. 7 (1966), 612-614. MR 35 #125. MR 0209223 (35:125)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0289454-4
Keywords: Complex quadratic field, ideal class group, least prime quadratic residue, extended Riemann Hypothesis
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society