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Communicated by E.M. Friedlander MSC: 16K50 16W50 16W60 16W70 a b s t r a c t 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].

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Abstract. We show in this article that in many cases the subfields of a nondegenerate tame semiramified division algebra of prime power degree over a Henselian valued field are inertial field extensions of the center [Th. 2.3, Th. 2.10 and Prop. 2.13 ].

This paper deals with the structure of nicely semiramified valued division algebras. We prove that any defectless finite-dimensional central division algebra over a Henselian field E with an inertial maximal subfield and a totally ramified maximal subfield (not necessarily of radical type) (resp., split by inertial and totally ramified field extensions of E) is nicely semiramified