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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
DOI: https://doi.org/10.1090/S0002-9939-1995-1249886-X
MathSciNet review: 1249886
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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 [Enhancements On Off] (What's this?)

  • 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
  • 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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Article copyright: © Copyright 1995 American Mathematical Society