Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


On the ideal class group of real biquadratic fields

Author: Patrick J. Sime
Journal: Trans. Amer. Math. Soc. 347 (1995), 4855-4876
MSC: Primary 11R29; Secondary 11R16
MathSciNet review: 1333398
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We discuss the structure of the ideal class group of real biquadratic fields $ K$, concentrating on the case that the $ 4$-rank of the ideal class groups of the quadratic subfields of $ K$ is 0. In this case, we give estimates for the $ 4$-rank of the ideal class group of $ K$. As an example, let $ K = \mathbb{Q}(\sqrt p ,\sqrt {627} )$, where $ p$ is a prime satisfying certain congruence conditions. The $ 2$-primary part of the ideal class group of $ K$ is then isomorphic to $ {(\mathbb{Z}/4\mathbb{Z})^2},\mathbb{Z}/4\mathbb{Z} \times {(\mathbb{Z}/2\mathbb{Z})^2}$, or $ {(\mathbb{Z}/2\mathbb{Z})^4}$. Further, each of the above occurs infinitely often.

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

  • [Ha1] Helmut Hasse, Zur Geschlechtertheorie in quadratischen Zahlkörpern, J. Math. Soc. Japan 3 (1951), 45–51 (German). MR 0043828 (13,324e)
  • [Ha2] Helmut Hasse, Number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 229, Springer-Verlag, Berlin-New York, 1980. Translated from the third German edition and with a preface by Horst Günter Zimmer. MR 562104 (81c:12001b)
  • [He] G. Herglotz, Über einen Dirichletschen Satz, Math. Z. 12 (1922), 225-261.
  • [Hi] D. Hubert, Gesammelte Abhandlungen, Vol. I, Chelsea, New York, 1965.
  • [Ja] Gerald J. Janusz, Algebraic number fields, Academic Press [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Pure and Applied Mathematics, Vol. 55. MR 0366864 (51 #3110)
  • [Kub] Tomio Kubota, Über den bizyklischen biquadratischen Zahlkörper, Nagoya Math. J. 10 (1956), 65–85 (German). MR 0083009 (18,643e)
  • [Kur] Sigekatu Kuroda, Über den Dirichletschen Körper, J. Fac. Sci. Imp. Univ. Tokyo. Sect. I. 4 (1943), 383–406 (German). MR 0021031 (9,12f)
  • [Ma] Daniel A. Marcus, Number fields, Springer-Verlag, New York-Heidelberg, 1977. Universitext. MR 0457396 (56 #15601)
  • [Si] P. Sime, On the ideal class groups of real biquadratic fields, Ph.D. Thesis, University of Maryland, College Park, 1992.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11R29, 11R16

Retrieve articles in all journals with MSC: 11R29, 11R16

Additional Information

PII: S 0002-9947(1995)1333398-3
Article copyright: © Copyright 1995 American Mathematical Society

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia