A geometric model of an arbitrary real closed field

Abstract

In the study of Hilbert’s 17th problem, orderings of a real field k are of importance. By the Artin-Schreier theorem the study of such orderings amounts to considering real closures of k. The aim of this talk is to construct a universal model of an arbitrary real closed field. To this end, we construct, in terms of Nash functions, all real closures of the rational function fields k = Q(ΛT ), where ΛT = (Λt : t ∈ T ) and T 6= ∅ is a system of any number of variables. This suffices to achieve our purpose, because any real closed field R is order-preserving isomorphic to a real closure of some field Q(ΛT ). If T = ∅, then Q(ΛT ) = Q, and the above is obvious. We also give a characterization of any Archimedean field in terms of fields of Nash functions.