Kummer subfields of tame division algebras
type: Item
instance of: scholarly article
title: Kummer subfields of tame division algebras
zbMATH Open document ID: 0840.16013
DOI: 10.1016/S0021-8693(05)80015-8
author: Patrick Morandi
author: B. A. Sethuraman
published in: Journal of Algebra
publication date: 1996-06-25T00:00:00Z
review text: The division algebras \(D\) considered in this paper are finite-dimensional over their center \(F\) and endowed with a (Schilling) valuation whose restriction to \(F\) is Henselian. It is further assumed that \(D=S\otimes T\) for some central division subalgebras \(S\), \(T\), where \(S\) is split by some inertial extension of \(F\) and \(T\) is totally ramified over \(F\) (the cases \(S=F\) or \(T=F\) are allowed) and that the residue characteristic does not divide the degree of \(D\). Under these hypotheses, the authors provide a homological description of the subfields of \(D\) which are Kummer (Galois) extensions of \(F\), generalizing results obtained by \textit{S. A. Amitsur} and the reviewer for certain division algebras of twisted formal power series [Isr. J. Math. 50, 114-144 (1985; Zbl 0564.16015)]. As an application of this description, they show that for any prime \(p\) and any \(n>m\geq 2\) the generic division algebra of degree \(p^n\) and exponent \(p^m\) remains a noncrossed product under any scalar extension of degree prime to \(p\). When \(D\) is a tensor product of two quaternion algebras and the residue field of \(F\) contains a primitive 4-th root of unity, they also provide a necessary and sufficient condition for \(D\) to be cyclic.
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 12J10
zbMATH DE Number: 763206
zbMATH Keywords: Schilling valuations
zbMATH Keywords: division algebras
zbMATH Keywords: central division subalgebras
zbMATH Keywords: generic division algebras
zbMATH Keywords: scalar extensions
zbMATH Keywords: tensor products
zbMATH Keywords: quaternion algebras
reviewed by: Jean-Pierre Tignol
MaRDI profile type: Publication
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full work available at URL: s0021-8693(05)80015-8
OpenAlex ID: W2019265281
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description: scientific article; zbMATH DE number 763206