Is it possible to show that there is a simple formula, preferably existential, that characterizes a nonstandard model of the ring of integers among the elements of $\prod_p \mathbb{F}_p(t)/\mathcal{U}$ where $\mathcal{U}$ is a nonprincipal ultrafilter over the prime numbers?

Thank you

definablesubset etc. – Emil Jeřábek Feb 2 '12 at 14:11