A Note on Existentially Closed Difference Fields with Algebraically Closed Fixed Field

We point out that the theory of difference fields with algebraically closed fixed field has no model companion. By a difference field we mean a field K equipped with an automorphism σ. It is well-known ([?]) that the class of existentially closed difference fields is an elementary class (ACFA), and moreover all completions are unstable. The fixed field (set of a ∈ K such that σ(a) = a) is responsible for the instability as it is pseudofinite. So in connection with an attempt to find stable nontrivial difference fields, one might consider the class of difference fields with algebraically closed fixed field and look for a model companion. Here we point out that this is impossible. We prove: Theorem 1 Let T be the theory of algebraically closed difference fields (K,+, ·,−, 0, 1, σ) with algebraically closed fixed field. Then T has no model companion. Equivalently the class of existentially closed models of T is not elementary. Thanks to Zoé Chatzidakis for some helpful comments, and to Dugald Macpherson and Kanat Kudaibergenov for some preliminary discussions. Let T be as in the assumptions of the theorem. Lemma 2 Let (K, σ) be an existentially closed model of T , and let a ∈ K be nonzero. Then there is n ≥ 1 and nonzero x ∈ K such that σ(x)/x = a.