Answer to a Question of Hung and Tiep on Conductors of Cyclotomic Integers
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908749129580544 |
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| author | Herbig, Christopher |
| author_facet | Herbig, Christopher |
| contents | In Question 5.2 of [5], Hung and Tiep asked the following: If $α$ is a sum of $k$ complex roots of unity and $\mathbb{Q}_{c(α)}$ is the smallest cyclotomic field containing $α$, is it true that $|\mathbb{Q}_{c(α)}:\mathbb{Q}(α)| \leq k$? 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 §6, we bound the growth of $|\mathbb{Q}_{c(α)}:\mathbb{Q}(α)|$ as a function of $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16732 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Answer to a Question of Hung and Tiep on Conductors of Cyclotomic Integers Herbig, Christopher Number Theory Group Theory In Question 5.2 of [5], Hung and Tiep asked the following: If $α$ is a sum of $k$ complex roots of unity and $\mathbb{Q}_{c(α)}$ is the smallest cyclotomic field containing $α$, is it true that $|\mathbb{Q}_{c(α)}:\mathbb{Q}(α)| \leq k$? 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 §6, we bound the growth of $|\mathbb{Q}_{c(α)}:\mathbb{Q}(α)|$ as a function of $k$. |
| title | Answer to a Question of Hung and Tiep on Conductors of Cyclotomic Integers |
| topic | Number Theory Group Theory |
| url | https://arxiv.org/abs/2508.16732 |