Elementary statements over large algebraic fields

Abstract:

We prove here the following theorems: A. If k is a denumerable Hilbertian field then for almost all $({\sigma _1}, \ldots ,{\sigma _e}) \in \mathcal {G}{({k_s}/k)^e}$ the fixed field of $\{ {\sigma _1}, \ldots ,{\sigma _e}\} ,{k_s}({\sigma _1}, \ldots ,{\sigma _e})$, has the following property: For any non-void absolutely irreducible variety V defined over ${k_s}({\sigma _1}, \ldots ,{\sigma _e})$ the set of points of V rational over K is not empty. B. If E is an elementary statement about fields then the measure of the set of $\sigma \in \mathcal {G}(\tilde Q/Q)$ (Q is the field of rational numbers) for which E holds in $\tilde Q(\sigma )$ is equal to the Dirichlet density of the set of primes p for which E holds in the field ${F_p}$ of p elements.