Baer ordered *-fields of the first kind
A skew field K with an involution \(a\mapsto a^*\) (we shall shortly say a *-field) of characteristic \(\neq 2\) will be called of the first kind, if all of its central elements are symmetric \((=\) self-adjoint). It is well known that the dimension of such a field over its center is \(4^ n\) for some natural number n. In this paper, the author considers *-fields with a ``Baer ordering''. Such an ordering is given by a set P of symmetric elements with the following properties: (i) P contains 1 but not 0, (ii) \(P+P\subseteq P\), (iii) \(bPb^*\subseteq P\) for all \(b\neq 0\), (iv) \(P\cap -P=\emptyset\), (v) every symmetric element different from 0 belongs to P or -P. In this paper the author constructs Baer ordered *- fields of the first kind of dimension \(4^ n\) over their center for every natural number n. The construction is explicit and relies on earlier work of \textit{L. H. Rowen} [cf.: Isr. J. Math. 29, 285-301 (1978; Zbl 0392.16011)].
scientific article; zbMATH DE number 4097573
scientific article