Theorem: If the extension $E/F$ is finite and Galois, then $E/F$ is normal and separable.
Proof: Let $f(x)\in F[x]$ irreducible with a root $\alpha\in E $ and $\sigma_1(\alpha),\dots,\sigma_r(\alpha)$ the orbit of $\alpha$ under $G(E/F).$
The polynomial $\prod_{i=1}^{r}(x-\sigma_i(\alpha))$ is separable and irreducible in $F[x].$
Therefore $\prod_{i=1}^{r}(x-\sigma_i(\alpha))$ splits over $E.$
Thus $E/F$ is normal and separable. ◼︎
Why $\prod_{i=1}^{r}(x-\sigma_i(\alpha))$ is separable and irreducible in $F[x]?$ If we know that it is irreducible, then we just need to see that every root $\sigma_i(\alpha)$ is different, i.e. $\sigma_i\neq\sigma_j$ for every $i\neq j.$ How?