2009/11/13 by Ming-chang Kang, Kang, Ming-chang
Computer Science · Mathematics · #13A50 #14E08 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.0911.2521
openalex publication_date 2009/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \bmC(V)G is retract \bmC-rational. Theorem 3. If G is a finite group, then all the Sylow subgroups of G are cyclic if and only if \bmCα(M)G is retract \bmC-rational for all G-lattices M, for all short exact sequences α: 0 → \bmC× → Mα → M → 0. Because the unramified Brauer group of a retract \bmC-rational field is trivial, Theorem 2 and Theorem 3 generalize previous results of Bogomolov and Barge respectively (see Theorem \reft5.9 and Theorem \reft6.1).