Realizable Classes and Embedding Problems
Cindy (Sin Yi) Tsang1
1Yau 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
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
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