Is there a matrix $A \in \mathbb{Q}^{n \times n}$ such that $P(A)=\mathbf{0}$, where $P$ is a monic polynomial with rational coefficients?

Question

Given a monic polynomial of degree $k \leq n$ with rational coefficents, can I always find a matrix $A \in \mathbb{Q}^{n \times n}$ that is a root of $P$?

What I have already tried:

• Using the Cayley-Hamilton theorem, without success.
• I know that the minimal polynomial of $A$ (if $A$ exists) must divide $P$.
• I also tried to relate rational roots (if any exists) of $P$ with the existence of $A$ (by using the rational root theorem), but I don't know how to make any progress from there.

Answer 1

The companion matrix of a polynomial $p$ has characteristic polynomial $p$.