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- Title
CLASSES DE STEINITZ D'EXTENSIONS NON ABÉLIENNES À GROUPE DE GALOIS D'ORDRE 16 OU EXTRASPÉCIAL D'ORDRE 32 ET PROBLÈME DE PLONGEMENT.
- Authors
SBEITY, FARAH; SODAÏGUI, BOUCHAÏB
- Abstract
Let k be a number field and Cl(k) its class group. Let Γ be a nonabelian group of order 16 or an extra-special group of order 32. Let Rm(k, Γ) 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 Γ. When Γ is the modular group of order 16, we assume that k contains a primitive 4th root of unity. In the present paper, we show that Rm(k, Γ) 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. We prove that for all c∈Cl(k), there exist a tame quadratic extension of k, with Steinitz class c, and which is embeddable in a tame Galois extension of k with Galois group isomorphic to Γ.
- Subjects
SET theory; ABELIAN groups; GALOIS theory; GROUP theory; EMBEDDINGS (Mathematics); NUMBER theory; MODULAR groups; ISOMORPHISM (Mathematics)
- Publication
International Journal of Number Theory, 2010, Vol 6, Issue 8, p1769
- ISSN
1793-0421
- Publication type
Article
- DOI
10.1142/S1793042110003794