Galois groups and rational solutions of $p(X) = A$

TL;DR

This paper extends previous work on matrix equations over rationals by establishing conditions for solutions based on Galois theory and polynomial factorization.

Contribution

It provides necessary and sufficient conditions for the existence of rational solutions to polynomial matrix equations, generalizing earlier results.

Findings

Conditions for $f(p(f))$ to have a degree $n$ factor over $b5.

Extension of Reams' theorem to broader polynomial matrices.

Characterization of solutions via Galois group properties.

Abstract

We extend Theorem 1 of R. Reams, A Galois approach to m-th roots of matrices with rational entries, LAA 258 (1997), 187-194. 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 $\mathbb{Q}.$