LINEAR RELATIONS BETWEEN THREE CONJUGATE ALGEBRAIC NUMBERS OF LOW DEGREE

LINEAR RELATIONS BETWEEN THREE CONJUGATE ALGEBRAIC NUMBERS OF LOW DEGREE

Let α_i, α_j, α_k be three distinct algebraic conjugates of an algebraic number α of degree d ≤ 8 over ℚ. In this paper by exploiting the properties of transitive permutation groups we determine all possible linear relations of the form aα_i + bα_j + cα_k = 0 with non-zero rationals a, b, c. We also provide the complete list of transitive groups which can occur as Galois groups for the minimal polynomial of α. For example, if d = 8, then we find that such relation is possible only if a² + b² = c² and the minimal polynomial of α over ℚ has Galois group isomorphic to D₄, the dihedral group of order 8. These findings extend the research of Dubickas and Jankauskas who investigated the relation aα_i + bα_j + cα_k = 0 for d ≤ 8 when a = b = c or a = b = -c.