Simple Linear Relations Between Conjugate Algebraic Numbers of Low Degree

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 ischolar_mains 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.