Journal of the Ramanujan Mathematical Society

Volume 30, Issue 2, June 2015 pp. 219–235.

Simple linear relations between conjugate algebraic numbers of low degree

Authors: Arturas Dubickas and Jonas Jankauskas
Author institution:Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania

Summary: We consider the linear equations α1 = α2 + α3 and α1 + α2 + α3 = 0 in conjugates of an algebraic number α of degree d ≤ 8 over Q. We prove that solutions to those equations exist only in the case d = 6 (except for the trivial solution of the second equation in cubic numbers with trace zero) and give explicit formulas for all possible minimal polynomials of such algebraic numbers. For instance, the first equation is solvable in roots of an irreducible sextic polynomial if and only if it is an irreducible polynomial of the form x6 + 2ax4 + a2x2 + b ∈ Q[x]. The proofs involve methods from linear algebra, Galois theory and some combinatorial arguments.

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