$E/F$ is a finite extension. Prove that $E/F$ is a normal extension $\iff$ $E$ is a splitting field for some polynomial $f\in F[X]$.
An extension $E/F$ is called normal if it is algebraic and if $\alpha \in E$ then minimal polynomial of $\alpha$ splits in $E$.
My try:
Let $\{a_1,a_2,...a_n\}$ be a basis of $E/F$. Let $f_i$ be the minimal polynomial of $a_i$ over $F$. Then $f_i$ splits in $E$. Consider $f=f_1f_2\cdots f_n$. Then $f$ splits in $E$. Since $a_i$ are the roots of $f$ then $E$ is the splitting field of $E$ over $F$.
Conversely, let $\alpha \in E$ and $g$ be the minimal polynomial of $\alpha $ over $F$. To show that $g$ splits in $E$
How to show this? Any help