Kummer subfields of tame division algebras over Henselian fields

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 DD over a Henselian valued field EE to have Kummer subfields when the characteristic of the residue field E¯ of EE does not divide the degree of DD. We prove also that if DD is a semiramified division algebra of degree nn [resp., of prime power degree prpr] over EE such that char(E¯) does not divide nn and rk(ΓD/ΓE)≥3 [resp., p≠char(E¯) and p3p3 divides exp(ΓD/ΓE)], then DD is non-cyclic [resp., DD is not an elementary abelian crossed product].