On places and preorderings of ternary fields
The author has made the remarkable discovery that the theories of Witt rings and of spaces of orderings [in the sense of \textit{M. Marshall}, Can. J. Math. 31, 320--330 (1979; Zbl 0412.10012)] can be developed for arbitrary ternary fields, see the author [``Spaces of orderings and Witt rings of planar ternary rings'', J. Pure Appl. Algebra 58, No. 2, 169--180 (1989; Zbl 0679.10017)]. The set \(Q(T)\) of non-zero ``generalized squares'' of a ternary field \(T\) plays an important rôle in this theory. For example, \(T\) admits an ordering if and only if \(P_0(T)\), the set of all sums of elements of \(Q(T)\), does not contain the element \(-1\) [see the author, J. Geometry 31, No.1/2, 100--113 (1988; Zbl 0642.06010) and ``Über Präordnungen von Ternärkörpern'', Abh. Math. Semin. Univ. Hamb. 58, 5--13 (1988; Zbl 0645.06010)]. An ordering \(P\) of \(T\) and a place \(\pi: T\to T'\cup \{\infty \}\) are said to be compatible if \(P^{\pi}\setminus \{0,\infty\}\) is an ordering of \(T'\). By result (2.5), the place \(\pi\) is compatible with some ordering of \(T\) if and only if \(Q(T)^{\pi}\setminus \{0,\infty\}\) is contained in some preordering of \(T'\). The orderings compatible with a given valuation of a field have been described by Baer and Krull, and this description has been extended to skew fields by \textit{A. Tschimmel} [``Über Anordnungsräume von Schiefkörpern'' (Dissertation, Münster 1981)]. The author obtains the following result for ternary fields \(T\): the orderings of \(T\) compatible with a given place \(\pi: T\to T'\cup \{\infty\}\) are in bijective (explicitly given) correspondence with the pairs \((P,sg)\), where \(P\subseteq P_ 0(T)^{\pi}\setminus \{0,\infty\}\) is an ordering of \(T'\) and \(sg\) is a character of the factor group \(T^*/P_ 0(T)({T'}^*)^{\pi^{-1}}\). A preordering \(F\) of \(T\) is a fan if every preordering \(sg\) satisfying \(sg(-1)=-1\) and \(sg(F)=1\) is already an ordering of \(T\). With this definition, many characterizations of fans of fields carry over. Moreover, the author generalizes a result of \textit{E. Becker} and \textit{E. Köpping} [Abh. Math. Semin. Univ. Hamb. 46, 143--177 (1977; Zbl 0365.12011)]: a non-trivial fan is not contained in any weakly archimedean ordering. Finally, a suitable definition of the stability degree of a preordering allows to establish the local stability formula of \textit{L. Bröcker} [Math. Ann. 210, 233--256 (1974; Zbl 0273.13018)] for ternary fields, and to characterize the SAP-preorderings as those preorderings of stability degree at most 1.
scientific article; zbMATH DE number 4059541