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.

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