The AMS website will be down for maintenance on May 23 between 6:00am - 8:00am EDT. For questions please contact AMS Customer Service at or (800) 321-4267 (U.S. & Canada), (401) 455-4000 (Worldwide).


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)



A lower bound for the class numbers of abelian algebraic number fields with odd degree

Author: Mao Hua Le
Journal: Proc. Amer. Math. Soc. 123 (1995), 1347-1350
MSC: Primary 11R29; Secondary 11M20, 11R20, 11R42
MathSciNet review: 1249886
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\Delta _K},{h_K},{R_K}$ denote the discriminant, the class number, and the regulator of the Abelian algebraic number field $ K = \mathbb{Q}(\alpha )$ with degree d, respectively. In this note we prove that if $ d > 1,2\nmid d$, and the defining polynomial of $ \alpha $ has exactly $ {r_1}$ real zeros and $ {r_2}$ pairs of complex zeros, then $ {h_K} > w\sqrt {\vert{\Delta _K}\vert} /{2^{{r_1}}}{(2\pi )^{{r_2}}}33{R_K}\log 4\vert{\Delta _K}\vert$, where w is the number of roots of unity in K.

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

  • [1] Pierre Barrucand, John Loxton, and H. C. Williams, Some explicit upper bounds on the class number and regulator of a cubic field with negative discriminant, Pacific J. Math. 128 (1987), no. 2, 209–222. MR 888515
  • [2] Jing Run Chen and Tian Ze Wang, On the distribution of zeros of Dirichlet 𝐿-functions, Sichuan Daxue Xuebao 26 (1989), no. Special Issue, 145–155 (Chinese, with English summary). MR 1059696
  • [3] E. Hecke, Vorlesung über die Theorie der algebraischen Zahlen, Akademische Verlagsgesellschaft, Leipzig, 1923.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11R29, 11M20, 11R20, 11R42

Retrieve articles in all journals with MSC: 11R29, 11M20, 11R20, 11R42

Additional Information

Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society