Classes de Steinitz d'extensions galoisiennes à groupe de Galois un $2$-groupe

Abstract

Let $k$ be a number field and $Cl(k)$ its class group. Let $\Gamma$ be a non trivial finite $2$-group. Let $R_m(k, \Gamma)$ be the subset of $Cl(k)$ consisting of those classes which are realizable as Steinitz classes of tame Galois extensions of $k$ with Galois group isomorphic to $\Gamma$. In the present article, we show that $R_m(k,\Gamma)$ is the full group $Cl(k)$, if the class number of $k$ is odd. We study an embedding problem connected with Steinitz classes in the perspective of studying realizable Galois module classes, when $\Gamma$ is defined by certain central non-split group extensions, examples of which are certain groups of order $32$ or $64$. For such groups $\Gamma$, We prove that for all $c\in Cl(k)$, there exist a tame quadratic extension of $k$, with Steinitz class $c$, and which is embeddable in a~Galois extension of $k$ with Galois group isomorphic to $\Gamma$.