  Let $K=\mathbb{Q}(\sqrt[4]{pd^{2}})$ be a real pure quartic number field and
$k=\mathbb{Q}(\sqrt{p})$ its real quadratic subfield, where $p\equiv 5\pmod 8$
is a prime integer and $d$ an odd square-free integer coprime to $p$. In this
work, we calculate $r_2(K)$, the $2$-rank of the class group of $K$, in terms
of the number of prime divisors of $d$ that decompose or remain inert in
$\mathbb{Q}(\sqrt{p})$, then we will deduce forms of $d$ satisfying $r_2(K)=2$.
In the last case, the $4$-rank of the class group of $K$ is given too.
