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Let $p_1$ and $p_2$ be two primes such that $p_1\equiv p_2\equiv1 \pmod4$ andat least two of the three elements $\{(\frac{2}{p_1}), (\frac{2}{p_2}),(\frac{p_1}{p_2})\}$ are equal to -1. Put $i=\sqrt{-1}$, $d=2p_1p_2$ and $k=Q(\sqrt{d}, i)$. Let $k_2^{(1)}$ be the Hilbert 2-class field of $k$ and$k^{(*)}=Q(\sqrt{p_1},\sqrt{p_2},\sqrt 2, i)$ be its genus field. Let $C_{k,2}$denote the 2-part of the class group of $k$. The unramified abelian extensionsof $k$ are $K_1=k(\sqrt{p_1})$, $K_2=k(\sqrt{p_2})$, $K_3=k(\sqrt{2})$ and$k^{(*)}$. Our goal is to study the capitulation problem of the 2-classes of$k$ in these four extensions.

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