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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0643733-3
PII: S 0002-9939(1982)0643733-3
Keywords: Class number, cyclotomic field
Article copyright: © Copyright 1982 American Mathematical Society