Linear relations of four conjugates of an algebraic number

Abstract We characterize all algebraic numbers $\alpha $ of degree $d\in \{4,5,6,7\}$ for which there exist four distinct algebraic conjugates $\alpha _1$ , $\alpha _2$ , $\alpha _3$ , and $\alpha _4$ of $\alpha $ satisfying the relation $\alpha _{1}+\alpha _{2}=\alpha _{3}+\alpha _{4}$ . In particular, we prove that an algebraic number $\alpha $ of degree 6 satisfies this relation with $\alpha _{1}+\alpha _{2}\notin \mathbb {Q}$ if and only if $\alpha $ is the sum of a quadratic and a cubic algebraic number. Moreover, we describe all possible Galois groups of the normal closure of $\mathbb {Q}(\alpha )$ for such algebraic numbers $\alpha $ . We also consider similar relations $\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}=0$ and $\alpha _{1}+\alpha _{2}+\alpha _{3}=\alpha _{4}$ for algebraic numbers of degree up to 7.