Minimal Mahler measures for generators of some fields

Abstract

We prove that for each odd integer $d \geq 3$ there are infinitely many number fields $K$ of degree $d$ such that each generator $\alpha$ of $K$ has Mahler measure greater than or equal to $d^{-d}|\Delta\_K|^{\frac{d+1}{d(2d-2)}}$, where $\Delta\_K$ 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 $d \geq 3$. We also show that for each $d \geq 2$ and any $\varepsilon>0$, there exist infinitely many number fields $K$ of degree $d$ such that every algebraic integer generator $\alpha$ of $K$ has Mahler measure greater than $(1-\varepsilon)|\Delta\_K|^{{1}/{d}}$. On the other hand, every such field $K$ contains an algebraic integer generator $\alpha$ with Mahler measure smaller that $|\Delta\_K|^{{1}/{d}}$. This generalizes the corresponding bounds recently established by Eldredge and Petersen for $d=3$.