Let
p
p
be an odd prime.
h
1
(
p
)
{h_1}(p)
is the first factor of the class number of field
Q
(
ζ
p
)
Q({\zeta _p})
. We proved that
\[
h
1
(
p
)
⩽
{
2
p
(
p
−
1
8
(
2
l
/
2
+
1
)
4
/
l
)
(
p
−
1
)
/
4
,
if
l
is even,
2
p
(
p
−
1
8
(
2
l
−
1
)
2
/
l
)
(
p
−
1
)
/
4
,
if
l
is odd
.
{h_1}(p) \leqslant \left \{ \begin {gathered} 2p{\left ( {\frac {{p - 1}} {{8{{({2^{l/2}} + 1)}^{4/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is even,}} \hfill \\ 2p{\left ( {\frac {{p - 1}} {{8{{({2^l} - 1)}^{2/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is odd}}{\text {.}} \hfill \\ \end {gathered} \right .
\]
From that we obtain
h
1
(
p
)
⩽
2
p
(
(
p
−
1
)
/
31.997158
…
)
(
p
−
1
)
/
4
{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
(
ζ
2
n
)
Q({\zeta _{{2^n}}})
and
Q
(
ζ
p
n
)
(
n
⩾
2
)
Q({\zeta _{{p^n}}})(n \geqslant 2)
, we get the similar results.