Realizable Classes and Embedding Problems
Cindy (Sin Yi) Tsang1
1 Yau Mathematical Sciences Center Tsinghua University Haidian District Beijing 100084, P.R. China
Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 647-680.

Let K be a number field and denote by 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 𝒪 h is a locally free 𝒪 K G-module by a classical theorem of E. Noether and it defines a class in the locally free class group Cl(𝒪 K G) of 𝒪 K G. We denote by R(𝒪 K G) 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(𝒪 K G) is connected to the study of embedding problems when G is abelian.

Soit K un corps de nombres et soit 𝒪 K son anneau des entiers. Soit G un groupe fini et soit K h une K-algèbre galoisienne de groupe G. Si K h /K est modérée, son anneau des entiers 𝒪 h est un 𝒪 K G-module localement libre d’après un théorème classique d’E. Noether et définit une classe dans le groupe des classes Cl(𝒪 K G) des 𝒪 K G-modules localement libres. On note R(𝒪 K G) l’ensemble de toutes ces classes. En combinant les travaux de L.R. McCulloh et J. Brinkhuis, on prouve que la structure de R(𝒪 K G) est liée à l’étude de problèmes de plongement lorsque G est abélien.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.995
Classification: 11R04, 11R32, 11R33
Keywords: Galois module, rings of integers, realizable classes, embedding problems

Cindy (Sin Yi) Tsang 1

1 Yau Mathematical Sciences Center Tsinghua University Haidian District Beijing 100084, P.R. China
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Cindy (Sin Yi) Tsang. Realizable Classes and Embedding Problems. Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 647-680. doi : 10.5802/jtnb.995. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.995/

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