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 D over a Henselian valued field E to have Kummer subfields when the characteristic of the residue field (E) over bar 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) over bar) does not divide n and rk(Gamma(D)/Gamma(E)) >= 3 [resp., p not equal char((E) over bar) and p(3) divides exp (Gamma(D)/Gamma(E))], then D is non-cyclic [resp., D is not an elementary abelian crossed product]. (C) 2009 Elsevier B.V. All rights reserved