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 ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$, ${\alpha }_4$ 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.