SCHWEITZER 2017 Problem 3

For every algebraic integer $\alpha$ define its positive degree $\text{deg}^+(\alpha)$ to be the minimal $k\in\mathbb{N}$ for which there exists a $k\times k$ matrix with non-negative integer entries with eigenvalue $\alpha$ . Prove that for any $n\in\mathbb{N}$ , every algebraic integer $\alpha$ with degree $n$ satisfies $\text{deg}^+(\alpha)\le 2n$ .