Exercise 5.2.2

Proof. Let $F\subset L$ a normal extension. Let $\alpha \in L$. Its minimal polynomial $f\in F[x]$ is irreducible, thus this polynomial splits completely over $F$ by definition of a normal extension.

Conversely, suppose that every $\alpha \in L$ is such that its minimal polynomial splits completely over $F$.

Let $g\in F[x]$ any irreducible polynomial, and $\alpha$ a root of $g$ in $L$. Then $g$ is the minimal polynomial of $\alpha$ over $L$. So $g$ splits completely over $L$ by hypothesis. Hence every irreducible polynomial $g$ which has a root in $L$ splits completely over $L$. So the extension $F\subset L$ is normal. □