Glasnik Matematicki, Vol. 60, No. 2 (2025), 229-242. \( \)

LINEAR RELATIONS BETWEEN THREE ALGEBRAIC CONJUGATES OF DEGREE TWICE A PRIME

Paulius Virbalas

Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
e-mail:paulius.virbalas@mif.vu.lt

Abstract.   In this paper, we show that there is no irreducible polynomial \(f(x)\) of degree \(2p\) (\(p\geq5\) is a prime number) over \({\mathbb Q}\) whose three distinct roots sum up to zero. This extends some earlier results on linear relations between three algebraic numbers. In particular, let \(d\) be the smallest positive integer not a multiple of \(3\), for which there exists an irreducible polynomial \(f(x)\) of degree \(d\) whose three distinct roots add up to zero. In 2015, Dubickas and Jankauskas found that \(10\leq d \leq 20\). As a corollary, we show that it is either \(d=16\) or \(d=20\).

2020 Mathematics Subject Classification.   11R32, 11F05, 12F10, 20B35

Key words and phrases.   Linear relations between polynomial roots, non-trivial additive relations between algebraic conjugates, transitive permutation groups.

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https://doi.org/10.3336/gm.60.2.04

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