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Let  K  = ℚ(    p   d   24   )$\begin{array}{} \displaystyle (\sqrt[4]{pd^{2}}) \end{array}$    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.

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