Definability in Global Fields and Finitely Generated Fields

Abstract

For algebraically closed fields, one cannot define characteristic zero by means of a single sentence in the language of rings. However, for global fields (i.e., finite separable extensions of Q or F_p[t]) characteristic zero is definable. In other words, there is a sentence in the language of rings which is true in a global field precisely when that global field has characteristic zero. In fact there are many subsets of the set of global fields which have a first-order definition and they correspond to the arithmetically definable subsets of the natural numbers. It turns out that characteristic zero is also definable for infinite finitely generated fields. The characterization of all subsets of infinite finitely generated fields is still an open problem.