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Alessandro Cobbe
Steinitz classes of some abelian and nonabelian extensions of even degree
Journal de théorie des nombres de Bordeaux, 22 no. 3 (2010), p. 607-628, doi: 10.5802/jtnb.735
Article PDF | Reviews MR 2769334 | Zbl 1267.11112

Résumé - Abstract

The Steinitz class of a number field extension $K/k$ is an ideal class in the ring of integers $\mathcal{O}_k$ of $k$, which, together with the degree $[K:k]$ of the extension determines the $\mathcal{O}_k$-module structure of $\mathcal{O}_K$. We denote by $\mathrm{R}_t(k,G)$ the set of classes which are Steinitz classes of a tamely ramified $G$-extension of $k$. We will say that those classes are realizable for the group $G$; it is conjectured that the set of realizable classes is always a group.

In this paper we will develop some of the ideas contained in [7] to obtain some results in the case of groups of even order. In particular we show that to study the realizable Steinitz classes for abelian groups, it is enough to consider the case of cyclic groups of $2$-power degree.

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