Weak \(*\)-orderings on \(*\)-fields
Let \(D\) be a division ring with center \(F\) and an involution \(*\). In this paper the author studies a special class of Baer-orderings, so called weak \(*\)-orderings, on \(D\). One motivation for this is the fact that \(*\)- fields which are finite dimensional over their centers only admit \(*\)- orderings if they are trivial. However, the author provides nontrivial examples of \(*\)-fields that are finite dimensional over their centers and have a weak \(*\)-ordering \(P\). The conditions under which those exist are described by Theorem 1.3: i) If \(*\) is of the first kind, \((D,x)\) is trivial. ii) If \(*\) is of the second kind, \([D:F]\) is odd. If the subfield \(k\subset D\) consists of symmetric elements only, then \((k,P\cap k)\) is an ordered field. To prove this result in \S4 the author uses the quite common tool of studying the valuation theory of Baer ordered \(*\)-fields and the connections between Baer orderings of the residue class field first. The main result of \S2 is Theorem 2.8 which states the existence of nontrivial valuations which are semicompatible with a nonarchimedean Baer ordering \(P\) on \(D\); in fact the order \(*\)-valuation \(v\) is nontrivial and \(P\) induces an ordering \(\overline{P}\) on \(\overline{D}_ v\). In Corollary 3.4 the author shows how a Baer ordering can be lifted from the residue class field \(\overline{D}\) to \(D\). In \S4 he proves a result analogous to 3.4 for weakly \(*\)-ordered fields (Theorem 4.7). In Theorem 4.9 he shows that if \([D:F]\) is odd, \((D,*)\) admits a weak \(*\)-ordering iff it admits a Baer ordering, which in particular implies that there exist nontrivial Baer ordered \(*\)-fields.
scientific article; zbMATH DE number 495543