  We prove a differential analog of a theorem of Chevalley on extending
homomorphisms for rings with commuting derivations, generalizing a theorem of
Kac. As a corollary, we establish that, under suitable hypotheses, the image of
a differential scheme under a finite morphism is a constructible set. We also
obtain a new algebraic characterization of differentially closed fields. We
show that similar results hold for differentially closed fields that are
saturated, in the sense of model theory. In characteristic p > 0, we obtain
related results and establish a differential Nullstellensatz. Analogs of these
theorems for difference fields are also considered.
