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

2025/08/22 by Herbig, Christopher
#FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.16732

Abstract

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

Citations

• Classifying minimal vanishing sums of roots of unity
• On vanishing sums for roots of unity
• Approximate formulas for some functions of prime numbers

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