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On the first factor of the class number of a cyclotomic field


Author: Ke Qin Feng
Journal: Proc. Amer. Math. Soc. 84 (1982), 479-482
MSC: Primary 12A50; Secondary 12A35
DOI: https://doi.org/10.1090/S0002-9939-1982-0643733-3
MathSciNet review: 643733
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Abstract: Let $ p$ be an odd prime. $ {h_1}(p)$ is the first factor of the class number of field $ Q({\zeta _p})$. We proved that

$\displaystyle {h_1}(p) \leqslant \left\{ \begin{gathered}2p{\left( {\frac{{p - ... ...\quad {\text{if }}l\;{\text{is odd}}{\text{.}} \hfill \\ \end{gathered} \right.$

From that we obtain $ {h_1}(p) \leqslant 2p{((p - 1)/31.997158 \ldots )^{(p - 1)/4}}$ which is better than Carlitz's and Metsänkyla's results. For the fields $ Q({\zeta _{{2^n}}})$ and $ Q({\zeta _{{p^n}}})(n \geqslant 2)$, we get the similar results.


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

  • [1] L. Carlitz, A generalization of Maillet's determinant and a bound for the first factor of the class number, Proc. Amer. Math. Soc. 12 (1961), 256-261. MR 0121354 (22:12093)
  • [2] E. E. Kummer, Bestimmung der Anzahl nicht Äquivalentar Klassen für die aus $ \lambda $-ten Wurzeln der Einheit gebildeten complexen Zählen und die ideale Faktoren derselben, J. Reine Angew. Math. 40 (1850), 93-116.
  • [3] R. Long, Algebraic number theory, Dekker, New York, 1977. MR 0469888 (57:9668)
  • [4] T. Metsänkyla, Class number and $ \mu $-invariants of cyclotomic fields, Proc. Amer. Math. Soc. 43 (1974), 199-200.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0643733-3
Keywords: Class number, cyclotomic field
Article copyright: © Copyright 1982 American Mathematical Society

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