Rational solutions of the matrix equation p (X) = A

We extend Theorem 1 of R. Reams, A Galois approach to m-th roots of matrices with rational entries, LAA, 258: 187-194, 1997. Let p () be any polynomial over Q, and let A Mₙ (Q) have irreducible characteristic polynomial f () with degree n. We provide necessary and sufficient conditions for the existence of a solution X Mₙ (Q) of the polynomial matrix equation p (X) = A. Specifically, we find necessary and sufficient conditions for f (p () ) to have a factor of degree n over Q.