No three algebraic conjugates of degree sixteen sum to zero

2026/08/04 by Žygimantas Baronėnas, Paulius Drungilas, Jonas Jankauskas
Mathematics · #math.NT #msc:11R04 #msc:11R09 #msc:11R32 #msc:12F10 #msc:20D20

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arxiv created 2026/08/04 · arxiv updated 2026/08/05

Abstract

Let d be the smallest positive integer, not a multiple of 3, for which there exists an algebraic number \al of degree d over ℚ whose three algebraic conjugates add to zero. We prove that d=20. This is derived from the following result: for any linear relation ∑_j=1^d a_j \al_j=0 with coefficients a_j∈ℤ among the conjugates \al_j of an algebraic number of degree d=p^m, where p is a prime number, m ≥ 1, the sum ∑_j=1a_j is divisible by p. If d=2p^m, p≥ 3 and ∑_j=1^d|a_d| < p, then ∑_j=1a_j is an even number.

Citations

• Linear relations of four conjugates of an algebraic number
• Orbits of Sylow subgroups of finite permutation groups
• A Census of Small Transitive Groups and Vertex-Transitive Graphs
• Practical graph isomorphism, II