Witt rings and orderings of skew fields

Abstract

While ordered skew fields have been known for a long time, they have received very little attention. One purpose of this paper is to show that studying orderings and associated structures is one way of obtaining information about skew fields which are infinite dimensional over their centers. All of our work is restricted to skew fields of characteristic not equal to 2. Of course, ordered skew fields have characteristic zero since they induce orderings on their centers. The first example of a noncommutative ordered field appears to be due to D. Hilbert in “Grundlagen der Geometrie” 115, $331. The example follows a proof of the fact that if a skew lield D has an archimedean ordering (every positive element of D is less than some rational integer), then D is commutative [15, $321. Another early result of a general nature was proved by A. A. Albert [I 1, based on earlier work of L. Dickson. Albert showed that an ordered skew field which is algebraic over its center must be commutative. We shall generalize this result in the next section. In 1952, T. Szele [28 ] extended to skew fields the basic results of Artin and Schreier on sums of squares [2,3] ( sums of products of squares for skew fields) and a criterion for extension of orderings due to Serre 1271. The second and third sections of our paper are devoted to extending these results and others to “orderings of higher level” in skew fields. Orderings of higher level were recently defined by E. Becker in [4] for commutative fields and appear to be of great importance in extending results for quadratic forms to forms of higher degree. In particular, 2”th powers take the place of squares in the Artin-Schreier theory. In fact, in (5 1 he has extended this to arbitrary even powers, but the valuation theory needed exceeds that which we have developed in Section 3 for skew fields. Section 2 covers the extension to skew