If a is algebraic over F and a root of some polynomial over F, then the conjugates of a are also roots

Let F be a field, E an extension of F, and let \theta_1, \theta_2 \in E be algebraic over F. Suppose further that \theta_1 and \theta_2 are conjugate. If p(x) \in F[x] has \theta_1 as a root, then \theta_2 is also a root.

If \theta_1 is a root of p(x), then the minimal polynomial s(x) of \theta_1 over F divides p(x) in F[x]. Since \theta_1 and \theta_2 are conjugate, \theta_2 is also a root of s(x), and hence of p(x).

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