Necessary and sufficient conditions are given for the class number
h
K
i
{h_{{K_i}}}
of a pure field
K
=
Q
(
m
1
/
p
i
)
(
for
i
=
1
,
2
)
K = Q({m^{1/{p^i}}}){\text { }}({\text {for }}i = 1,2)
to be divisible by
p
r
{p^r}
for a given positive integer
r
r
and prime
p
p
. Moreover the divisibility of
h
K
i
{h_{{K_i}}}
by
p
p
is linked with the
p
p
-rank of the class group of the
K
(
ς
)
K(\varsigma )
and prime divisors of
m
m
, where
ς
\varsigma
is a primitive
p
p
th root of unity. Finally we prove in an easy fashion that for a given odd prime
p
p
and any natural number
t
t
there exist infinitely many non-Galois algebraic number fields (in fact pure fields) of degree
p
i
(
i
=
1
,
2
)
{p^i}(i = 1,2)
over
Q
Q
whose class numbers are all divisible by
p
t
{p^t}
.