Simple Linear Relations Between Conjugate Algebraic Numbers of Low Degree

Abstract

We consider the linear equations a1 = a2 + a3 and a1 + a2 + a3 = 0 in conjugates of an algebraic number a of degree d :s 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 x 6 + 2ax4 +a2x 2 +b E (Q)[x]. The proofs involve methods from linear algebra, Galois theory and some combinatorial arguments.