This is an attempt to extend the current full fledged random matrix theory to fields of positive characteristics. So here is a possible setup for the problem: Let $A_{n,p}$ be an $n \times n$ matrix with entries iid taking values uniformly in $F_p$. Then one should be able to find its eigenvalues together with multiplicities, which might lie in some finite extension of the field $F_p$. To ensure diagonalizability, one might even take $A_{n,p}$ to be symmetric or antisymmetric (I am not so sure if that guarantees diagonalizability in $F_p$ but I have no counterexamples either). Now the question is if we associate to each eigenvalue $\lambda$ the degree of its minimal polynomial $d(\lambda)$, then does the distribution of $d(\lambda)$ as $n$ goes to infinite converge to some law upon normalization (say maybe Gaussian)? I am very curious whether others have studied this problem before. Maybe it's completely trivial.

1$\begingroup$ Being symmetric isn't enough; for example, if p is congruent to 1 mod 4, let i denote a square root of 1. Then [[i 1][1 i]] squares to zero. $\endgroup$ – Qiaochu Yuan Apr 3 '10 at 2:21

2$\begingroup$ Anyway, section 1.10 of the second edition of Stanley's EC Chapter might be relevant: math.mit.edu/~rstan/ec/ch1.pdf $\endgroup$ – Qiaochu Yuan Apr 3 '10 at 2:28

$\begingroup$ Nice example! I guess one could still ask the question in the nonsymmetric case, as an analogy of the ginibre Ensemble in the complex case. $\endgroup$ – John Jiang Apr 3 '10 at 4:05
The survey article
Jason Fulman, Random matrix theory over finite fields, Bulletin of the AMS 39 (2002), 5185
and the references therein should answer your questions to the extent that the answers are currently known. See in particular Example 3 in Section 2.2. Roughly, the distribution of the degrees of the factors of the characteristic polynomial of a random matrix over a finite field is close to the distribution of the degrees of the factors of a random polynomial over the same finite field, which is close to the distribution of the cycle lengths of a random element of a symmetric group.

$\begingroup$ Thank you for the reference and pointer to the example. This is very helpful. $\endgroup$ – John Jiang Apr 3 '10 at 20:56