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Positive degree of algebraic integer at most 2n

Source: Miklós Schweitzer 2017, problem 3

January 13, 2018
Algebraic integereigenvaluenumber theory

Problem Statement

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