Sur une question de capitulation

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

RSESUM´E. Soient $p$ et $q$ deux nombres premiers tels que $p \equiv 1 \bmod 8,\,\, q \equiv -1\bmod 4$et $(\frac{\textstyle p}{\textstyle q}) = - 1$, $d = pq$, $i = \sqrt{-1}$, $\mathbf{k} = \mathbf{Q}(\sqrt{d},i)$, $\mathbf{k}^{(1)}_{2}$ le 2-corps de classes de Hilbert de $\mathbf{k}$, $\mathbf{k}^{(2)}_{2}$le 2-corps de classes de Hilbert de $\mathbf{k}^{(1)}_{2}$et $G_{2}$ le groupe de Galois de $\mathbf{k}^{(2)}_{2}/\mathbf{k}$. La 2-partie $C_{\mathbf{k}, 2}$du groupe de classes de $\mathbf{k}$est de type $(2,2)$, par suite $\mathbf{k}^{(1)}_{2}$contient trois extensions $\mathbf{K}_{i}/\mathbf{k},\,\,i = 1,\,2,\,3$. On s'intéresse au problème de capitulation des 2-classes de $\mathbf{k}$ dans $\mathbf{K}_{i},\,\,i = 1,\,2,\,3$, et à déterminer la structure de $G_{2}$.