Model-complete theories of e-free Ax fields

Abstract

This paper's goal is to determine which complete theories of perfect, e-free Ax fields are model-complete. A field K is e-free for a positive integer e if the Galois group g(KS∣K), where Ks is the separable closure of K, is an e-free, profinite group. A perfect field K is pseudo-algebraically closed if each nonvoid, absolutely irreducible variety defined over K has a K-rational point. A perfect, pseudo-algebraically closed field is called an Ax field. The main theorem isA complete theory of e-free Ax fields is model-complete if and only if its field of absolute numbers is e-free.The sufficiency of the latter condition is an easy consequence of a result of Moshe Jarden and Ursel Kiehne [10] and has been noted independently by A. Macintyre and K. McKenna and undoubtedly by others as well. Consequently the necessity of the latter condition is the interesting part of this paper.James Ax [3] initiated the investigation of 1-free Ax fields. He proved that these fields, which he called pseudo-finite fields, are precisely the infinite models of the theory of finite fields. He [3] also presented examples of perfect, 1-free fields which are not pseudo-algebraically closed and an example of a 1-free Ax field whose complete theory is not model-complete. Moshe Jarden [5] showed that the first examples are isolated cases in that almost all, perfect, 1-free, algebraic extensions of a denumerable, Hilbertian field are pseudo-algebraically closed. The results in this paper show that the second example is also an isolated case in that almost all complete theories of 1-free Ax fields are model-complete.