# Minimal Mahler measures for generators of some fields

### Artūras Dubickas

Vilnius University, Lithuania

## Abstract

We prove that for each odd integer $d≥3$ there are infinitely many number fields $K$ of degree $d$ such that each generator $α$ of $K$ has Mahler measure greater than or equal to $d_{−d}∣Δ_{K}∣_{d(2d−2)d+1}$, where $Δ_{K}$ is the discriminant of the field $K$. This, combined with an earlier result of Vaaler and Widmer for composite $d$, answers negatively a question of Ruppert raised in 1998 about ‘small’ algebraic generators for every $d≥3$. We also show that for each $d≥2$ and any $ε>0$, there exist infinitely many number fields $K$ of degree $d$ such that every algebraic integer generator $α$ of $K$ has Mahler measure greater than $(1−ε)∣Δ_{K}∣_{1/d}$. On the other hand, every such field $K$ contains an algebraic integer generator $α$ with Mahler measure smaller that $∣Δ_{K}∣_{1/d}$. This generalizes the corresponding bounds recently established by Eldredge and Petersen for $d=3$.

## Cite this article

Artūras Dubickas, Minimal Mahler measures for generators of some fields. Rev. Mat. Iberoam. 39 (2023), no. 1, pp. 269–282

DOI 10.4171/RMI/1331