Valuated division algebras and crossed products
type: Item
instance of: scholarly article
title: Valuated division algebras and crossed products
zbMATH Open document ID: 0615.16015
DOI: 10.1016/0021-8693(87)90127-X
author: Achim Kupferoth
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: The author's aim is to extend Amitsur's result on non-crossed products [\textit{S. A. Amitsur}, Isr. J. Math. 12, 408-420 (1972; Zbl 0248.16006)]. Let \(U=UD(K,n,t)\) be the generic division algebra of t \((\geq 2)\) \(n\times n\) matrices over a field K. Suppose that K has a rank 1 valuation, with residue class field \(\tilde K.\) He proves that if U is a crossed product, then so is \(\tilde U= UD(\tilde K,n,t)\); moreover if char\((\tilde K)\) does not divide n and U is a crossed product with a group G, then \(\tilde U\) is a crossed product with G. Combined with Amitsur's theorem this shows that if \(UD(Q,n,t)\) is a crossed product, then any division algebra D of dimension \(n^ 2\) over its centre is a crossed product (with the same group in case char\((D) \nmid n)\). The proof is by an analysis of crossed products using a result of Procesi on generic division algebras.
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16W20
zbMATH DE Number: 3995908
zbMATH Keywords: generic division algebra
zbMATH Keywords: valuation
zbMATH Keywords: residue class field
zbMATH Keywords: crossed products
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
cites work: On central division algebras
cites work: PI-algebras. An introduction
cites work: Q5602178
cites work: Noncrossed Products of Small Exponent
cites work: Q5620935
cites work: Q5796078
cites work: Q5511421
description: scientific article; zbMATH DE number 3995908