Minimal Mahler measures for generators of some fields

Abstract

We prove that for each odd integer $d\ge 3$ 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 $d\ge 3$. We also show that for each $d\ge 2$ and any $\epsilon >0$, there exist infinitely many number fields $K$ of degree $d$ such that every algebraic integer generator $\alpha$ of $K$ has Mahler measure greater than $(1-\epsilon )|{\Delta }_K{|}^{1/d}$. 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 $d=3$.