Let $p$ and $q$ be prime numbers such that $p \equiv 1 \bmod 8,\,\, q \equiv -1 \bmod 4$ and $(\frac{\textstyle p}{\textstyle q}) = - 1$, $d = pq$, $\k = \Q(\sqrt{d},i)$, $\k^{(1)}_{2}$ be the 2-Hilbert class field of $\k$, $\k^{(2)}_{2}$ be the 2-Hilbert class field of $\k^{(1)}_{2}$ and $G_{2}$ be the Galois group of $\k^{(2)}_{2}/\k$. The 2-part $C_{\k,2}$ of the class group of $\k$ is of type $(2,2)$, so $\k_{2}^{(1)}$ contains three extensions $\K_{i}/\k,\,\,i = 1,\,2,\,3$. Our goal is to determine the group $C_{\k,2}$, to study the problem of capitulation of the 2-classes of $\k$ in $\K_{i},\,\,i = 1,\,2,\,3$ and to construct the 2-class field tower of $\k $.

\medskip\noindent {\bf R\'esum\'e.} \

Soient $p$ et $q$ deux nombres premiers tels que $p \equiv 1 \bmod 8,\,\, q \equiv -1\bmod 4$ et $(\frac{\textstyle p}{\textstyle q}) = - 1$, $d = pq$, $i = \sqrt{-1}$, $\k = \Q(\sqrt{d},i)$, $\k^{(1)}_{2}$ le 2-corps de classes de Hilbert de $\k$, $\k^{(2)}_{2}$ le 2-corps de classes de Hilbert de $\k^{(1)}_{2}$ et $G_{2}$ le groupe de Galois de $\k^{(2)}_{2}/\k$. La 2-partie $C_{\k, 2}$, du groupe de classes de $\k$ est de type $(2,2)$, par suite $\k^{(1)}_{2}$ contient trois extensions $\K_{i}/\k,\,\,i = 1,\,2,\,3$. On s'int\'eresse \`a d\'eterminer le groupe $C_{\k,2}$, \`a etudier la capitulation des 2-classes de $\k$ dans $\K_{i},\,\,i = 1,\,2,\,3$ et \`a la construction de la tour du 2-corps de classes de Hilbert de $\k $.