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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.

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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. –  Qiaochu Yuan Apr 3 '10 at 2:21
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Anyway, section 1.10 of the second edition of Stanley's EC Chapter might be relevant: math.mit.edu/~rstan/ec/ch1.pdf –  Qiaochu Yuan Apr 3 '10 at 2:28
    
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. –  John Jiang Apr 3 '10 at 4:05
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up vote 12 down vote accepted

The survey article

Jason Fulman, Random matrix theory over finite fields, Bulletin of the AMS 39 (2002), 51-85

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.

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