Splitting fields for characteristic polynomials of matrices with entries in a finite field

Let MnMn be the set of all n×nn×n matrices with entries in the finite field FqFq. Let X(A)X(A) be the degree of the splitting field of the characteristic polynomial of A  , and let μnμn be the average degree:μn=1|Mn|∑A∈MnX(A). A theorem of Reiner is used to prove that, as n→∞n→∞,μn=eBn/logn(1+o(1)), where B is an explicit constant.