On the 2-rank and 4-rank of the class group of some real pure quartic number fields

Let K = ℚ ( p d 2 4 ) $\begin{array}{} \displaystyle (\sqrt[4]{pd^{2}}) \end{array}$ be a real pure quartic number field and k = ℚ( p $\begin{array}{} \displaystyle \sqrt{p} \end{array}$ ) its real quadratic subfield, where p ≡ 5 (mod 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 ℚ( p $\begin{array}{} \displaystyle \sqrt{p} \end{array}$ ), 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.