The Elementary Theory of e-Free PAC Fields

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 20.3.3) 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 coeficients in K that are true in $\tilde{K}(\sigma)$ , for almost all σ∈G(K)^e (Section 20.5). 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 20.9.3): A sentence θ of ℒ(ring,O _K ) is true among the fields $\tilde{K}(\sigma)$ with probability equal to the probability that is true among the residue fields of K.