TITLE: Is there a geometric interpretation of $F_p,\ F_{p^n}$ and $\overline{F_p}?$
QUESTION [13 upvotes]: I have been doing some exercises about finite fields lately and I think I've obtained some understanding of what they are. What seems to be missing though is some kind of picture. Learning to work with complex numbers seems pretty much equivalent to learning what the field operations do on the complex plane geometrically. At least, it makes things much, much easier. Is there a way to imagine finite fields and their algebraic closure geometrically?
I was thinking that perhaps I should look at $F_p$ as the set $\mathbb C_p$ of $p$-th roots of $1$ in $\mathbb C.$ But it doesn't seem to work very well. The additive structure of $F_p$ translates well (via an isomorphism) to the multiplicative structure of $\mathbb C_p.$ But transporting the multiplicative structure $F_p$ to $\mathbb C_p$ via the same function gives something I can't really visualize. It's the operation $\star$ given by the following formula:
$$\left(\cos\frac{2k\pi}{p}+i\sin\frac{2k\pi}{p}\right)\star \left(\cos\frac{2l\pi}{p}+i\sin\frac{2l\pi}{p}\right)=\left(\cos\frac{2kl\pi}{p}+i\sin\frac{2kl\pi}{p}\right).$$
But even if it's the right way to do it, it still doesn't immediately give a way of seeing $F_{p^n}$ there too, let alone the algebraic closure $\overline{F_p}.$ And of course, it would be good to have an interpretation of $\overline{F_p}$ in which all $F_{p^n}$ for $n\in \mathbb N$ could be easily distinguished.
So my question is 

Are there any useful ways of interpreting the finite fields and their algebraic closures geometrically? If so what are they?

REPLY [7 votes]: My own idiosyncratic way of looking at these things: Remember that the multiplicative group of ${\mathbb{F}}_{p^n}$ is cyclic, so you could, if you wished, model this also as a group of roots of unity in $\mathbb{C}$. This helps tremendously in doing hand computations in the finite field, by the way, since once you find a generator (“primitive root”) and write down all its powers, you have a nice log table for quick multiplication. At any rate, these finite cyclic groups all fit together very nicely, so that the multiplicative group of the algebraic closure of your finite field of characteristic $p$ is (noncanonically) isomorphic to the group of all $m$-th roots of unity in $\mathbb{C}$ for $m$ ranging through the integers prime to $p$.
Edit: I originally said that the group in question was also isomorphic to ${\mathbb{Q}}_p/{\mathbb{Z}}_p$, and this was quite wrong.