TITLE: Algebraically closed fields of positive characteristic
QUESTION [47 upvotes]: I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far as intuition goes, I might as well add "...of characteristic 0" at the end of that.
I know the complex numbers from kindergarten algebra, so I have a fairly good idea of how at least one algebraically closed field of characteristic 0 looks and feels. And while I don't have nearly the same handle on the field of algebraic numbers, I can pretty much do arithmetic in it, so that's two examples.
But I've never (really) seen an algebraically closed field of characteristic p > 0! I can build one just fine, and if you put a gun to my head I could probably even do some arithmetic in it, but there's no intuition of the sort that you get with the complex numbers. So: does anyone know of an intuitive description of such a field, that it's possible to get a real sense of in the same way as C?

REPLY [8 votes]: 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.