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

TL;DR

This paper addresses a question about the degree of cyclotomic fields generated by sums of roots of unity, providing counterexamples, characterizations for small sums, and bounds on the degree growth.

Contribution

It answers negatively a question by Hung and Tiep, characterizes cyclotomic integers with small sums where the inequality fails, and bounds the degree growth relative to the sum size.

Findings

Counterexamples to the inequality for certain sums

Complete characterization for sums with k ≤ 4

Bounds on the degree growth as a function of k

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 $|\mathbb{Q}_{c(\alpha)}:\mathbb{Q}(\alpha)| \leq k$? We answer this question in the negative. Using known results on minimal vanishing sums, we also characterize all cyclotomic integers with $k \leq 4$ for which the inequality fails. In \S 6, we bound the growth of $|\mathbb{Q}_{c(\alpha)}:\mathbb{Q}(\alpha)|$ as a function of $k$.