Original scientific paper

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

Linear relations between three algebraic conjugates of degree twice a prime

Paulius Virbalas orcid.org/0009-0008-2717-0380 ; Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania

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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 \(d=20\).

Keywords

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

Hrčak ID:

342481

URI

https://hrcak.srce.hr/342481

Publication date:

28.5.2026.

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