## In any extension of QQ, an algebraic element and its complex conjugate are conjugates

Let $E$ be an extension of $\mathbb{Q}$ contained in $\mathbb{C}$. Suppose $\alpha \in E$ is algebraic over $\mathbb{Q}$ with minimal polynomial $p(x)$ and such that the complex conjugate $\overline{\alpha}$ of $\alpha$ is also in $E$. Prove that $\alpha$ and $\overline{\alpha}$ are conjugates in $E$.

Note that complex conjugation is an automorphism of $\mathbb{C}$. Thus $p(\overline{\alpha}) = \overline{p(\alpha)} = \overline{0} = 0$. Since $\overline{\alpha}$ is a root of the minimal polynomial $p(x)$, $\alpha$ and $\overline{\alpha}$ are conjugate in $E$.