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arXiv · 2312.02044

Small generators of abelian number fields

Martin Widmer
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Abstract

We show that for each abelian number field $K$ of sufficiently large degree $d$ there exists an element $\alpha\in K$ with $K=\IQ(\alpha)$ and absolute Weil height $H(\alpha)\ll_d |\Delta_K|^{1/2d}$ , where $\Delta_K$ 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 $1/2d$ is best-possible when $d$ is even.

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Martin Widmer. 2023-12-04. Small generators of abelian number fields. https://arxiv.org/abs/2312.02044

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