Uniform first-order definitions in finitely generated fields

Abstract

We prove that there is a first-order sentence in the language of rings that is true for all finitely generated fields of characteristic $0$ and false for all fields of characteristic greater than $0$. We also prove that for each $n\in \mathbb{N}$, there is a first-order formula ${\psi }_n(x_1,\dots ,x_n)$ that when interpreted in a finitely generated field $K$ is true for elements $x_1,\dots ,x_n\in K$ if and only if the elements are algebraically dependent over the prime field in $K$