Spectrally arbitrary, nonderogatory factorization over a general field

It is known that a nonsingular, nonscalar matrix A, over the complex field, may be factored as A=BC, in which the spectra of B and C are arbitrary, subject to detBdetC=detA, and that B and C may be taken to be nonderogatory. The purpose of this paper is to establish this result over a general field with at least four elements.