Kummer subfields of tame division algebras over Henselian valued fields

Abstract

By generalizing the method used by Tignol and Amitsur in [TA85], we determine necessary and sufficient conditions for an arbitrary tame central division algebra D over a Henselian valued field E to have Kummer subfields [Corollary 2.11 and Corollary 2.12]. We prove also that if D is a tame semiramified division algebra of prime power degree p^n over E such that p\neq char(\bar E) and rk(\Gamma_D/\Gamma_E)\geq 3 [resp., such that p\neq char(\bar E) and p^3 divides exp(\Gamma_D/\Gamma_E)], then D is non-cyclic [Proposition 3.1] [resp., D is not an elementary abelian crossed product [Proposition 3.2]].