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arXiv · 2111.14256

Totally real algebraic integers of arboreal height 2

George J. Schaeffer
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Abstract

In arXiv:1302.4423, Salez proved that every totally real algebraic integer is the eigenvalue of some tree. We define the "arboreal height" of a totally real algebraic integer $\lambda$ to be the minimal height of a rooted tree having $\lambda$ as an eigenvalue. In this paper, we prove several results about totally real algebraic integers of arboreal height $2$: We show that all real quadratic integers have arboreal height $\le 2$. We characterize the totally real cubic integers of arboreal height $2$. Finally, we prove that every totally irrational real number field is generated (as a ring over $\mathbb{Q}$) by some $\lambda$ of arboreal height $2$.

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George J. Schaeffer. 2021-11-28. Totally real algebraic integers of arboreal height 2. https://arxiv.org/abs/2111.14256

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