Valuations and Hermitian forms on skew fields

Abstract

Abstract. This paper is a survey of the literature on ways in which the concept of ordering can be extended to the setting of a division ring with involution and the main results for these extensions.

Signatures, sums of hermitian squares and positive cones on algebras with involution

“…Other notions of orderings have been introduced for division rings with involution, most notably Baer orderings, * -orderings and their variants and an extensive theory has been developed around them. Craven's surveys [13] and [14] provide more information on these topics. Without going into the details, the main difference in the definitions is that positive cones were developed to correspond to a pre-existing algebraic notion, namely signatures of hermitian forms (e.g.…”

Abstract theory of semiorderings

“…In this section we look at theorems which are motivated by theorems for real valuations on fields. The role of the value group will be taken by certain subgroups of G, and so is actually analogous to the value group modulo 2 for fields (and a slightly more complicated group for *-fields [10,11]). Our first result shows how to construct new spaces of semiorderings from old ones.…”

“…Marshall's theory was more closely related to what actually happens with Witt rings of fields than the work in [20] (and this was later put into a ring-theoretic context by Rosenberg and Kleinstein [18]). Since that time, Marshall's work has been used by numerous authors; the abstract approach has been shown to provide a common way of proving theorems for not just fields, but also semilocal rings [17] and *-fields [11]. In spite of all the work in this area, there are still major open questions, most notably those of realisability of abstract spaces and realisability of images of Witt rings.…”

“…A totally different occurrence of semiorderings begins with the work of R. Baer [2]. Baer orderings of a •-field give an example of a noncommutative generalisation of semiordering for which very little is known concerning the Witt groups that arise (see [11] for a discussion of various generalisations to *-fields). This more general context will be taken up briefly in Section 2.…”