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

Abstract

Let $p_1$ and $p_2$ be two primes such that $p_1\equiv p_2\equiv 1(\mathrm{mod}4)$ and at least two of the three elements $\{(\frac{2}{p_1}),(\frac{2}{p_2}),(\frac{p_1}{p_2})\}$ are equal to -1. Put $i=\sqrt{-1}$, $d=2p_1{p}_2$ and $k=Q(\sqrt{d},i)$. Let $k_2^{(1)}$ be the Hilbert 2-class field of $k$ and $k^{(\ast )}=Q(\sqrt{p_1},\sqrt{p_2},\sqrt{2},i)$ be its genus field. Let $C_{k,2}$ denote the 2-part of the class group of $k$. The unramified abelian 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.

6 pages, in French in Workshop International "Théorie des Nombres, Codes, Cryptographie et Systèmes de Communication", du 26 au 28 Avril 2012 01/2012