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

Abstract

We characterize all algebraic numbers $α$ of degree $8$ for which there exist four distinct algebraic conjugates $α_1$, $α_2$, $α_3$, $α_4$ of $α$ satisfying the linear relation $α_{1}=α_{2}+α_{3}+α_{4}$. Analogous characterization is obtained for the linear relation $α_{1}+α_{2}+α_{3}+α_{4}=0$. In particular, when an algebraic number $α$ of degree $8$ has a non-even minimal polynomial and possesses exactly six distinct linear relations of the form $α_{i_1}+α_{i_2}+α_{i_3}+α_{i_4}=0$, we prove that $α$ is a sum of a quadratic and a quartic algebraic number.