If $\text{char}(F)\nmid[E:F]$ then $E/F$ is separable

Question

Let $E/F$ be a finite field extension. Suppose $p=\text{char}(F),n=[E:F]$ and $p\nmid n$. Prove that $E/F$ is separable.

I don't really have an attempt just a few observarions like if $p=0$ is we're done. So suppose $p>0$. Of course $p$ is a prime and we have $n=ap+b$ with $b\ne 0$.

Answer 1

Let $\alpha\in E$. Then the degree of the minimal polynomial of $\alpha$ (let's call it $f$) is $[F(\alpha):F]$ which divides $n$ and hence not divisible by $p$. From here we can easily get that the derivative of $f$ is non-zero. Since $f$ is irreducible over $F$ and $0\leq deg(f')<deg(f)$ we conclude that $\gcd(f,f')=1$ and hence $f$ is separable.

Answer 2

Hint: It is enough to show that if $F=E(\alpha)$ and $p$ does not divide $[F:E]$, then the minimal polynomial of $\alpha$ is separable. What are the non-separable irreducible polynomials?