Linear relations of four conjugates of an algebraic number

Abstract

We characterize all algebraic numbers $α$ of degree $d\in\{4,5,6,7\}$ for which there exist four distinct algebraic conjugates $α_1$, $α_2$, $α_3$, $α_4$ of $α$ satisfying the relation $α_{1}+α_{2}=α_{3}+α_{4}$. In particular, we prove that an algebraic number $α$ of degree 6 satisfies this relation with $α_{1}+α_{2}\notin\mathbb{Q}$ if and only if $α$ is the sum of a quadratic and a cubic algebraic number. Moreover, we describe all possible Galois groups of the normal closure of $\mathbb{Q}(α)$ for such algebraic numbers $α$. We also consider similar relations $α_{1}+α_{2}+α_{3}+α_{4}=0$ and $α_{1}+α_{2}+α_{3}=α_{4}$ for algebraic numbers of degree up to 7.