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Cindy (Sin Yi) Tsang
Realizable Classes and Embedding Problems
Journal de théorie des nombres de Bordeaux, 29 no. 2 (2017), p. 647-680, doi: 10.5802/jtnb.995
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Class. Math.: 11R04, 11R32, 11R33
Keywords: Galois module, rings of integers, realizable classes, embedding problems

Résumé - Abstract

Let $K$ be a number field and denote by $\mathcal{O}_K$ its ring of integers. Let $G$ be a finite group and let $K_h$ be a Galois $K$-algebra with group $G$. If $K_h/K$ is tame, then its ring of integers $\mathcal{O}_h$ is a locally free $\mathcal{O}_KG$-module by a classical theorem of E. Noether and it defines a class in the locally free class group $\mathrm{Cl}(\mathcal{O}_KG)$ of $\mathcal{O}_KG$. We denote by $R(\mathcal{O}_KG)$ the set of all such classes. By combining the work of L.R. McCulloh and J. Brinkhuis, we shall prove that the structure of $R(\mathcal{O}_KG)$ is connected to the study of embedding problems when $G$ is abelian.

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