Retract rational fields

JOURNAL OF ALGEBRA

卷:349

Article

Kang, Ming-chang^1,2 

[1] Natl Taiwan Univ, Dept Math, Taipei 10764, Taiwan

[2] Natl Taiwan Univ, Taida Inst Math Sci, Taipei 10764, Taiwan

关键词: Noether's problem;    Rationality problem;    Retract rational;    Multiplicative group actions;   

DOI  :  10.1016/j.jalgebra.2011.10.024

来源: Elsevier

【 摘 要 】

Let k be an infinite field. The notion of retract k-rationality was introduced by Saltman in the study of Noether's problem and other rationality problems. We will investigate the retract rationality of a field in this paper. Theorem 1: Let k subset of K subset of L be fields. If K is retract k-rational and L is retract K-rational, then L is retract k-rational. Theorem 2: For any finite group G containing an abelian normal subgroup H such that G/H is a cyclic group, for any complex representation G -> GL(V), the fixed field C(V)(G) is retract C-rational. Theorem 3: If G is a finite group, then all the Sylow subgroups of G are cyclic if and only if C-alpha(M)(G) is retract C-rational for all G-lattices M, for all short exact sequences alpha : 0 -> C-x -> M-alpha -> M -> 0. Because the unramified Brauer group of a retract C-rational field is trivial. Theorems 2 and 3 generalize previous results of Bogomolov and Barge respectively (see Theorems 5.9 and 6.1). (C) 2011 Elsevier Inc. All rights reserved.

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