Small generators of abelian number fields

Abstract

We show that for each abelian number field K of sufficiently large degree d there exists an element α ∈ K {\alpha\in K} with K = ℚ ⁢ ( α ) {K=\mathbb{Q}(\alpha)} and absolute Weil height H ⁢ ( α ) ≪ d | Δ K | 1 2 ⁢ d {H(\alpha)\ll_{d}|\Delta_{K}|^{\frac{1}{2d}}} , where Δ K {\Delta_{K}} denotes the discriminant of K. This answers a question of Ruppert from 1998 in the case of abelian extensions of sufficiently large degree. We also show that the exponent 1 2 ⁢ d {\frac{1}{2d}} is best-possible when d is even.