Model Completeness of Valued Pac Fields

We present a theorem of Kollár on the density property of valued PAC fields and a theorem of Abraham Robinson on the model completeness of the theory of algebraically closed non-trivial valued fields. Then we prove that the theory T of non-trivial valued fields in an appropriate first order language has a model completion T̃ . The models of T̃ are non-trivial valued fields (K, v) that are ω-imperfect, ω-free, and PAC. MR Classification: 12E30 * Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation. Introduction A field K is said to be PAC (pseudo algebraically closed) if every absolutely irreducible variety V defined over K (i.e. a geometrically integral K-scheme) has a Krational point. Here and throughout the paper we use K̃ to denote a fixed algebraic closure of K. The notion of PAC fields has been introduced in [Ax68] (although not by this name) in connection with the decidability of the elementary theory of finite fields. Each countable Hilbertian field K has an abundance of separable algebraic extensions of K which are PAC. Indeed, for each positive integer e and for almost all σ ∈ Gal(K), the fixed field Ks(σ) is PAC [FrJ08, Thm. 18.6.1]. Here Ks is the separable algebraic closure of K, Gal(K) = Gal(Ks/K) is the absolute Galois group of K, “almost all” is meant in the sense of the Haar measure of Gal(K) with respect to its Krull topology, and Ks(σ) is the fixed field in Ks of the coordinates of σ = (σ1, . . . , σe). Chapter 11 of [FrJ08] gives an extensive treatment of PAC fields. In particular, it points out that if K is PAC, then V (K) is Zariski dense in V (K̃) for each absolutely irreducible variety V defined over K [FrJ08, p. 192, Prop. 11.1.1] and asks whether V (K) is even v-dense in V (K̃) for each valuation v of K̃ [FrJ05, Problem 11.5.4]. If this happens, we say that K has the density property. The latter problem goes back to [GeJ75, Problem 1], where the following theorem is proved: Let K be a countable Hilbertian field and e a positive integer. Then for every valuation v of K̃, for almost all σ ∈ Gal(K), and for every absolutely irreducible variety V defined over K, the set V (Ks(σ)) is v-dense in V (K̃) [GeJ75, Thm. 6.2]. Note that the order of the quantifiers “for every valuation v” and “for almost all σ ∈ Gal(K)” can not be exchanged without a substantial argument, because K̃ has in general uncountably many valuations. That argument is supplied in [FrJ76], where the “stability of PAC fields” is proved [FrJ76, Thm. 3.4]. As a result, it is proved that Ks(σ) has the density property for almost all σ ∈ Gal(K). For a general PAC fieldK and an arbitrary valuation v of K̃, Prestel proved thatK is v-dense in K̃ [FrJ08, p. 204, Prop. 11.5.3]. The proof is based on the observation that if f ∈ K[X] is a nonconstant separable polynomial and c ∈ K×, then f(X1)f(X2)−c is an