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(λ) be any polynomial over Q, and let A∈ Mₙ(Q) have irreducible characteristic polynomial f(λ) 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(λ)) to have a factor of degree n over Q.