Retract rational fields

Abstract

Let k be an infinite field. The notion of retract k-rationality was introduced by Saltman in the study of Noetherʼs 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 ( 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 α ( M ) G is retract C -rational for all G-lattices M, for all short exact sequences α : 0 → C × → M α → 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).