Sur la capitulation des 2-classes d'id\'eaux du corps Q(\sqrt{2p_1p_2}, i)

Let $p_1$ and $p_2$ be two primes such that $p_1\equiv p_2\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and at least two of the three elements ${(\frac{2}{p_1}), (\frac{2}{p_2}), (\frac{p_1}{p_2})}$areequalto-1.Put$i=\sqrt{-1}$,$d=2p_1p_2$and$k =Q(\sqrt{d}, i)$\mathrm{.}Let$k_2^{(1)}$betheHilbert2-classfieldof$k$and$k^{(*)}=Q(\sqrt{p_1},\sqrt{p_2},\sqrt 2, i)$beitsgenusfield\mathrm{.}Let$C_{k,2}$denotethe2-partoftheclassgroupof$k$. The unramified abelian extensions of $k$ are $K_1=k(\sqrt{p_1})$, $K_2=k(\sqrt{p_2})$, $K_3=k(\sqrt{2})$ and $k^{(\ast )}$. Our goal is to study the capitulation problem of the 2-classes of $k$ in these four extensions.