The Elementary Theory of e-free PAC Fields

Abstract

This chapter presents one of the highlights of this book, the study of the elementary theory of e-free PAC fields. We apply the elementary equivalence theorem for arbitrary PAC fields (Theorem 18.6) to the theory of perfect e-free PAC fields containing a fixed countable base field K. If K is finite and e =1 or K is Hilbertian and e ≥ 1, then this theory coincides with the theory of all sentences with coefficients in K that are true in

$$\tilde K\left( \sigma \right)$$

, for almost all σ ∉ G(K)^e (Section 18.5). In particular, if K is explicitly given with elimination theory, then this theory is recursively decidable. In the special case where K is a global field and e = 1 we prove a transfer theorem (Theorem 18.26): A sentence ϑ of ϕ (ring, O _K ) is true among the fields

$$\tilde K\left( \sigma \right)$$

with probability equal to the probability that ϑ is true among the residue fields of K. Finally we prove that the elementary theory of finite fields is recursively decidable (Theorem 18.31).