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

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 $\mathrm{\mid }{\mathbb{Q}}_{c(\alpha )}:\mathbb{Q}(\alpha )\mathrm{\mid }\le k$? 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 $\mathrm{\mid }{\mathbb{Q}}_{c(\alpha )}:\mathbb{Q}(\alpha )\mathrm{\mid }$ as a function of $k$.