If you have fields $\displaystyle F\leq E$ with $\displaystyle \alpha \in E$ algebraic over $\displaystyle F$, show that,

$\displaystyle F(\alpha)$ is the minimal field containing $\displaystyle F$ and $\displaystyle \alpha$.

(Note: $\displaystyle F(\alpha)=\phi_{\alpha}[F[x]]$-a simple extension).