Answer to a Question of Hung and Tiep on Conductors of Cyclotomic Integers

Abstract

In Question 5.2 of [5], Hung and Tiep asked the following: If $\alpha$ is a sum of $k$ complex roots of unity and ${\mathbb{Q}}_{c(\alpha )}$ is the smallest cyclotomic field containing $\alpha$, is it true that $|{\mathbb{Q}}_{c(\alpha )}:\mathbb{Q}(\alpha )|\le k$? We answer this question in the negative. Using known results on minimal vanishing sums, we also characterize all cyclotomic integers with $k\le 4$ for which the inequality fails. In §6, we bound the growth of $|{\mathbb{Q}}_{c(\alpha )}:\mathbb{Q}(\alpha )|$ as a function of $k$.

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• Problems on the conductor of finite group characters