A complex version of the baer-krull theorems

Abstract

The Baer-Krull theorems deal with the relationship between orderings of a valued field compatible with the valuation, and orderings of the residue class field. For these theorems it is necessary that the valuation ring should be convex with respect to the ordering. For a real field R, and an extension K _ R, the author defines SpecC(K/R) in terms of equivalence classes of embeddings of K over R into an algebraic closure C of K. This is done in such a way that when K is also real, the points of SpecC(K/R) correspond to the orderings of K over R. Given a point of SpecC(K/R), the author extends the definition of convexity to subsets of K (again this is the usual definition when K is real). Now let R be real, K be an extension of R, and B be a valuation ring in K. Let R be the residue class field of R\B. Suppose that R is a real subfield of K. The author studies relations between SpecC(K/R) and SpecC(K/R). In particular, it is shown that there is a lifting of each element of SpecC(K/R) to an element of SpecC(K/R), compatible with the valuation, and such that the lifting has the generalised convexity property. While a more elementary treatment of this result is possible if R = Q, for general R the proof involves model theory in a nontrivial way.