Pseudo Algebraically Closed Fields over Rings

We prove that for almost all σ ∈ G(Q)e the field Q̃(σ) has the following property: For each absolutely irreducible affine variety V of dimension r and each dominating separable rational map φ: V → Ar there exists a point a ∈ V (Q̃(σ)) such that φ(a) ∈ Zr. We then say that Q̃(σ) is PAC over Z. This is a stronger property then being PAC. Indeed we show that beside the fields Q̃(σ) other fields which are algebraic over Q and are known in the literature to be PAC are not PAC over Z. Introduction J. Ax observed in [Ax] that every nonprincipal ultraproduct K of finite fields has the following property, which later on Frey [Fre] called PAC: Every absolutely irreducible variety defined over K has a K-rational point. Ax asked in [Ax] whether there exists a PAC field which is algebraic over Q besides the algebraic closure Q̃ of Q. The first author [Ja1] gave a host of examples for such fields. Indeed, he proved that if e is a positive integer, then Q̃(σ) is PAC for almost all σ ∈ G(Q). Here G(Q) is the absolute Galois group of Q, ‘almost all’ is used in the sense of the Haar measure of G(Q), and Q̃(σ) is the fixed field in Q̃ of σ = (σ1, . . . , σe). Later on more examples of algebraic extensions of Q which are PAC were given. Thus, [FJ1] constructs a Galois extension N of Q which is PAC such that G(N/Q) is a direct product of symmetric groups. Recently Pop proved for the maximal totally real extension Qtr of Q that Qtr( √ −1) is PAC [Pop]. * Partially supported by a grant from the Israel Academy of Sciences Received August 22, 1993