It is a little unfair to pass this off as an answer to your question, but it also seems too interesting to be buried as a comment. (I can say that because I am citing other people's work!)
The comments following Ben's answer point out some nice senses in which ‘any’ calculation over the algebraic closure of $\mathbb F_p$ is really a calculation over a suitably large finite extension of $\mathbb F_p$, but I didn't see a precise statement in the comments. I think it is very much worthwhile to note that, not just algebraically-closed-positive-characteristic computations, but even algebraically-closed-characteristic-$0$ calculations can be viewed this way. For example, this is one way to prove that an injective, polynomial self-map of $\mathbb C^n$ is bijective.
See Serre's lovely article and Tao's lovely summary of it for more details on this point of view.
