## Functiones et Approximatio Commentarii Mathematici

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

Bouchaïb Sodaïgui

#### 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 $\Gamma$.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 48, Number 2 (2013), 183-196.

Dates
First available in Project Euclid: 18 June 2013

https://projecteuclid.org/euclid.facm/1371586155

Digital Object Identifier
doi:10.7169/facm/2013.48.2.2

Mathematical Reviews number (MathSciNet)
MR3100139

Zentralblatt MATH identifier
1288.11100

#### Citation

Sodaïgui, Bouchaïb. Classes de Steinitz d'extensions galoisiennes à groupe de Galois un $2$-groupe. Funct. Approx. Comment. Math. 48 (2013), no. 2, 183--196. doi:10.7169/facm/2013.48.2.2. https://projecteuclid.org/euclid.facm/1371586155