How random is the characteristic polynomial of a random matrix ?

Every monic, degree n polynomial in F q [ x ;] is the characteristic polynomial of at least one n × n matrix (with entries in the finite field F q ), but they do not appear with equal frequency. There is no a priori reason that the characteristic polynomial of a typical matrix should resemble a typical monic degree n polynomial. Nevertheless, we prove a precise version of the following heuristic statement: ‘Excepting its small factors, the characteristic polynomial of a random matrix is random.’