Defining \(k\) in \(G(k)\)
type: Item
instance of: scholarly article
title: Defining \(k\) in \(G(k)\)
zbMATH Open document ID: 0935.20031
author: Gerhard Röhrle
author: Katrin Tent
author: Linus Kramer
published in: Journal of Algebra
publication date: 2000-02-13T00:00:00Z
review text: The authors are concerned with an interesting problem, namely, how the field of definition \(k\) of a \(K\)-isotropic absolutely almost simple \(k\)-group \(G\) ``lives'' in the group \(G(k)\) of \(k\)-rational points. The construction, inspired by the fundamental work of Borel and Tits, uses model theory. It is proved that if \(G\) is a \(k\)-simple \(K\)-isotropic group such that \(G(k)\) has finite Morley rank, then \(k\) is either algebraically closed or real closed; if \(G\) is absolutely simple \(K\)-isotropic, then \(k\) is algebraically closed. This answers a question of \textit{A. Borovik} and \textit{A. Nesin} [Groups of finite Morley rank, Oxford Clarendon Press (1994; Zbl 0816.20001)] under the assumption that \(G\) is \(k\)-isotropic.
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 20A15
Mathematics Subject Classification ID: 03C07
zbMATH DE Number: 1309250
zbMATH Keywords: reductive linear algebraic groups
zbMATH Keywords: field of definition
zbMATH Keywords: model theory
zbMATH Keywords: almost simple groups
zbMATH Keywords: groups of rational points
zbMATH Keywords: groups of finite Morley rank
MaRDI profile type: Publication
full work available at URL: jabr.1998.7771
OpenAlex ID: W4205362194
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cites work: Complements à l'article: ''Groupes reductifs''
cites work: Homomorphismes ''abstraits'' de groupes algébriques simples
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cites work: Q5648549
DOI: 10.1006/JABR.1998.7771
description: scientific article; zbMATH DE number 1309250