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

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 \[ {h_1}(p) \leqslant \left \{ \begin {gathered} 2p{\left ( {\frac {{p - 1}} {{8{{({2^{l/2}} + 1)}^{4/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is even,}} \hfill \\ 2p{\left ( {\frac {{p - 1}} {{8{{({2^l} - 1)}^{2/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is odd}}{\text {.}} \hfill \\ \end {gathered} \right .\] 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.