Is there any natural way of defining a structure in $\mathbb{C}$ (or some domain $D$ over complex n-vectors, or similar) so that it resembles a finite field $\mathbb{F}_{p^n}$?
For example, I am hoping something like the following can be done:
A set of vectors in $\mathbb{C}^n$ together with two certain operations is isomorphic to $\mathbb{F}_{p^n}$, and the operations have something to do with complex vector arithmetic.
I am not sure if such simple structures can be rich enough to capture the details of $\mathbb{F}_{p^n}$, though.