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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A lower bound for the class numbers of abelian algebraic number fields with odd degree
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by Mao Hua Le PDF
Proc. Amer. Math. Soc. 123 (1995), 1347-1350 Request permission

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 ${h_K} > w\sqrt {|{\Delta _K}|} /{2^{{r_1}}}{(2\pi )^{{r_2}}}33{R_K}\log 4|{\Delta _K}|$, where w is the number of roots of unity in K.
References
  • 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, DOI 10.2140/pjm.1987.128.209
  • 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
  • E. Hecke, Vorlesung über die Theorie der algebraischen Zahlen, Akademische Verlagsgesellschaft, Leipzig, 1923.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 1347-1350
  • MSC: Primary 11R29; Secondary 11M20, 11R20, 11R42
  • DOI: https://doi.org/10.1090/S0002-9939-1995-1249886-X
  • MathSciNet review: 1249886