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Found problems: 1

2017 Miklós Schweitzer, 3
Tags: algebraic integer , eigenvalue , number theory
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 $\text{deg}^+(\alpha)\le 2n$.