Linear relations of four conjugates of an algebraic number

2025/07/18 by Žygimantas Baronėnas, Baronėnas, Ž., Paulius Drungilas +3 · 1 citation
Computer Science · Mathematics · #11R04 #11R32 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2507.13831

openalex publication_date 2025/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We characterize all algebraic numbers α of degree d∈\4,5,6,7\ for which there exist four distinct algebraic conjugates α₁, α₂, α₃, α₄ of α satisfying the relation α₁+α₂=α₃+α₄. In particular, we prove that an algebraic number α of degree 6 satisfies this relation with α₁+α₂∉ℚ if and only if α is the sum of a quadratic and a cubic algebraic number. Moreover, we describe all possible Galois groups of the normal closure of ℚ(α) for such algebraic numbers α. We also consider similar relations α₁+α₂+α₃+α₄=0 and α₁+α₂+α₃=α₄ for algebraic numbers of degree up to 7.

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