On the height of polynomials that split completely over a fixed number field

Thian Tromp

On the height of polynomials that split completely over a fixed number field

Number Theory 2025-09-16 v1

Authors: Thian Tromp

Abstract

Let $K/\mathbb{Q}$ be a finite extension. We prove that the minimal height of polynomials of degree $n$ of which all roots are in $K^\times$ increases exponentially in $n$ . We determine the implied constant exactly for totally real $K$ and $K$ equal to $\mathbb{Q}(\sqrt{-1})$ or $\mathbb{Q}(\sqrt{-3})$ .

Keywords

height of polynomials number fields complete splitting algebraic number theory

Cite

@article{arxiv.2509.12064,
  title  = {On the height of polynomials that split completely over a fixed number field},
  author = {Thian Tromp},
  journal= {arXiv preprint arXiv:2509.12064},
  year   = {2025}
}

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