Functiones et Approximatio Commentarii Mathematici

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

Bouchaïb Sodaïgui

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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

Permanent link to this document
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

Subjects
Primary: 11R33: Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10]

Keywords
$2$-groups Galois module structure ring of integers realizable Steinitz classes embedding problem central extensions local symbols Brauer group

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


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