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arXiv · 2606.06222

No three algebraic conjugates of degree sixteen sum to zero

Žygimantas Baronėnas·Paulius Drungilas·Jonas Jankauskas
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Abstract

Let $d$ be the smallest positive integer, not divisible by $3$, for which there exists an algebraic number over $\mathbb{Q}$ of degree $d$ whose some three algebraic conjugates sum to zero. Employing the classification of vertex-transitive graphs on 16 vertices of degree 6, we prove that $d\neq 16$. This, combined with results obtained by Dubickas, Smyth and Stong \cite{DubickasSmyth2006}, Dubickas and Jankauskas \cite{DubickasJankauskas2015} and Virbalas \cite{Virbalas2025a}, implies that $d=20$.

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Žygimantas Baronėnas, Paulius Drungilas, Jonas Jankauskas. 2026-06-04. No three algebraic conjugates of degree sixteen sum to zero. https://arxiv.org/abs/2606.06222

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