Linear relations of four conjugates of an algebraic number of degree eight

Abstract

We characterize all algebraic numbers $\alpha$ of degree $8$ for which there exist four distinct algebraic conjugates ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$, ${\alpha }_4$ of $\alpha$ satisfying the linear relation ${\alpha }_1={\alpha }_2+{\alpha }_3+{\alpha }_4$. Analogous characterization is obtained for the linear relation ${\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_4=0$. In particular, when an algebraic number $\alpha$ of degree $8$ has a non-even minimal polynomial and possesses exactly six distinct linear relations of the form ${\alpha }_{i_1}+{\alpha }_{i_2}+{\alpha }_{i_3}+{\alpha }_{i_4}=0$, we prove that $\alpha$ is a sum of a quadratic and a quartic algebraic number.