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 $p(x)$.