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

Abstract

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 $p(\lambda)$ be any polynomial over $\mathbb{Q}$, and let $A\in M_n(\mathbb{Q})$ have irreducible characteristic polynomial $f(\lambda)$ with degree $n$. We provide necessary and sufficient conditions for the existence of a solution $X\in M_n(\mathbb{Q})$ of the polynomial matrix equation $p(X) = A.$ Specifically, we    find necessary and sufficient conditions for $f(p(\lambda))$ to have a factor of degree $n$ over $\mathbb{Q}.$