Submission: 2007, Jun 10
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 $rk(\Gamma_D/\Gamma_F)\geq 3$ [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]].
2000 Mathematics Subject Classification: 16K50, 16W50, 16W60 and 16W70.
Keywords and Phrases: Generalized crossed products, (Graded) Brauer group, Valued division algebras, Henselization, Graded division algebras, Kummer graded field extensions.
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