Topological division algebras without corresponding topological affine plane
type: Item
instance of: scholarly article
title: Topological division algebras without corresponding topological affine plane
zbMATH Open document ID: 0780.51009
DOI: 10.1007/BF02941624
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1993-08-31T00:00:00Z
review text: Let \(D\) be a topological (non-associative) division algebra. If \(D\) is alternative, then \(D\) gives rise to a topological projective plane, see \textit{H. Salzmann} [Math. Z. 67, 436-466 (1957; Zbl 0078.341)]. The author shows that this is not true for arbitrary (non-alternative) division algebras. More precisely, he proves the following result. Let \(K\) be a field of infinite transcendency degree over its prime field, and let \(D\) be a division algebra of finite dimension over its center \(K\). If \(D\) is not alternative, then \(D\) can be topologized in such a way that \(D\) becomes a topological division algebra which does not belong to any topological projective (or affine) plane. If \(K\) is real closed, then one can take the product topology on \(D\) derived from any nondiscrete valuation topology of \(K\) which is distinct from the order topology.
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 51A25
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 17A35
zbMATH DE Number: 404313
zbMATH Keywords: (topological) division algebra
zbMATH Keywords: topological projective plane
zbMATH Keywords: affine plane
author: Eberhard F. Eisele
reviewed by: Theo Grundhöfer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 404313