Retract rational fields

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 CG(V) 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 CαG(M) is retract C-rational for all G-lattices M, for all short exact sequences α:0→C×→Mα→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).