标题： 代数数的四个共轭的线性关系 显示英文标题

标题： Linear relations of four conjugates of an algebraic number

摘要： 我们表征所有次数为$d\in\{4,5,6,7\}$的代数数$\alpha$，其中存在四个不同的代数共轭$\alpha_1$、$\alpha_2$、$\alpha_3$、$\alpha_4$，它们是$\alpha$的共轭，并满足关系$\alpha_{1}+\alpha_{2}=\alpha_{3}+\alpha_{4}$。 具体而言，我们证明了 6 次代数数 $\alpha$ 满足以下关系： $\alpha_{1}+\alpha_{2}\notin\mathbb{Q}$ 当且仅当 $\alpha$ 是一个二次代数数与一个三次代数数之和。 此外，我们描述了此类代数数 $\alpha$ 的 $\mathbb{Q}(\alpha)$ 的正常闭包的所有可能的伽罗瓦群。 对于最高 7 次的代数数，我们还考虑了类似的关系： $\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}=0$ 和 $\alpha_{1}+\alpha_{2}+\alpha_{3}=\alpha_{4}$。 显示英文摘要

摘要： 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.