Kummer subfields of tame division algebras over Henselian valued fields

TL;DR

This paper characterizes the existence of Kummer subfields in tame division algebras over Henselian fields and explores conditions under which such algebras are non-cyclic or not elementary abelian crossed products.

Contribution

It generalizes previous methods to provide necessary and sufficient conditions for Kummer subfields in tame division algebras over Henselian fields and analyzes their structural properties.

Findings

Identifies conditions for Kummer subfields in tame division algebras.

Shows certain tame division algebras are non-cyclic under specific invariants.

Demonstrates some tame division algebras are not elementary abelian crossed products.

Abstract

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_E)\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]].