Totally real algebraic integers of arboreal height 2

TL;DR

This paper investigates totally real algebraic integers with arboreal height 2, establishing bounds for quadratic integers, characterizing cubic integers, and showing that all totally irrational real number fields can be generated by such integers.

Contribution

It provides new characterizations and bounds for totally real algebraic integers of arboreal height 2, extending Salez's work to specific degrees and fields.

Findings

All real quadratic integers have arboreal height ≤ 2.

Characterization of totally real cubic integers of arboreal height 2.

Every totally irrational real number field is generated by an algebraic integer of arboreal height 2.

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 $2$.