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

摘要

Let M_n be the set of all n × n matrices with entries in the finite field F_q. Let X(A) be the degree of the splitting field of the characteristic polynomial of A, and let μ_n be the average degree: μ_n=1/|M_n|∑ A∈M_n X(A). A theorem of Reiner is used to prove that, as n → ∞, μ_n=e~(B(n/log n(1+o(1)))) where B is an explicit constant.