Defining \(k\) in \(G(k)\)
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.
scientific article; zbMATH DE number 1309250