Kummer subfields of tame division algebras over Henselian fields

Abstract

By generalizing the method used by Tignol and Amitsur in [J.-P. Tignol, S.A. Amitsur, Kummer subfields of Malcev–Neumann division algebras, Israel Journal of Math. 50 (1985), 114–144], we determine necessary and sufficient conditions for an arbitrary central division algebra D over a Henselian valued field E to have Kummer subfields when the characteristic of the residue field E ¯ of E does not divide the degree of D . We prove also that if D is a semiramified division algebra of degree n [resp., of prime power degree p r ] over E such that char ( E ¯ ) does not divide n and rk ( Γ D / Γ E ) ≥ 3 [resp., p ≠ char ( E ¯ ) and p 3 divides exp ( Γ D / Γ E ) ], then D is non-cyclic [resp., D is not an elementary abelian crossed product].