In Question 5.2, Hung and Tiep (Eur J Math 9, 2023) asked the following: If <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> is a sum of k complex roots of unity and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {Q}_{c(\alpha )}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Q</mi> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> is the smallest cyclotomic field containing <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, is it true that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$|\mathbb {Q}_{c(\alpha )}:\mathbb {Q}(\alpha )| \le k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="double-struck">Q</mi> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mrow> <mo>:</mo> <mi mathvariant="double-struck">Q</mi> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mo>≤</mo> <mi>k</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation>? We answer this question in the negative, and in §4, we shall bound the growth of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$|\mathbb {Q}_{c(\alpha )}:\mathbb {Q}(\alpha )|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="double-struck">Q</mi> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mrow> <mo>:</mo> <mi mathvariant="double-struck">Q</mi> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> as a function of k using known results on minimal vanishing sums.

Answer to a question of Hung and Tiep on conductors of cyclotomic integers

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发表日期：

2026年1月6日

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Christopher Herbig

摘要

In Question 5.2, Hung and Tiep (Eur J Math 9, 2023) asked the following: If $\alpha $ $α$ is a sum of k complex roots of unity and $\mathbb {Q}_{c(\alpha )}$ $Q_{c (α)}$ is the smallest cyclotomic field containing $\alpha $ $α$ , is it true that $|\mathbb {Q}_{c(\alpha )}:\mathbb {Q}(\alpha )| \le k$ $| Q_{c (α)} : Q (α) | \leq k$ ? We answer this question in the negative, and in §4, we shall bound the growth of $|\mathbb {Q}_{c(\alpha )}:\mathbb {Q}(\alpha )|$ $| Q_{c (α)} : Q (α) |$ as a function of k using known results on minimal vanishing sums.