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.