Retract rational fields
Journal
Journal of Algebra
Journal Volume
349
Journal Issue
1
Pages
22-37
Date Issued
2012
Author(s)
Abstract
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⊂ K⊂ 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{double-struck}(V)G is retract C{double-struck}-rational. Theorem 3: If G is a finite group, then all the Sylow subgroups of G are cyclic if and only if C{double-struck}α(M)G is retract C-rational for all G-lattices M, for all short exact sequences α:0→C{double-struck}×→Mα→M→0. Because the unramified Brauer group of a retract C{double-struck}-rational field is trivial, Theorems 2 and 3 generalize previous results of Bogomolov and Barge respectively (see Theorems 5.9 and 6.1). © 2011 Elsevier Inc.
Type
journal article