If $\operatorname{char} K =p$ and $[F:K]$ is finite and not divisible by $p$, then $F$ is separable over $K$

Question

Consider the following question,

If $\operatorname{char} K =p$ and $[F:K]$ is finite and not divisible by $p$, then prove that $F$ is separable over $K$.

I am really sorry but I am not able to use the given information to solve the problem. I am not able to deduce anything using $p$ doesn't divides $[F:K]$ .

It is my humble request to you to kindly give some hints.

I am following Abstarct Algebra by Joseph Gallian.

Answer 1

Assume that $F/K$ is not separable.

There is $a\in F$ with $K$-minimal polynomial $f\in K[x]$ such that $K(a)/K$ is not separable, ie. $f'(a)=0$.

But then $f(a)=f'(a)=0$ means that $\gcd(f,f')\ne 1$, if also $\gcd(f,f')\ne f$ we get that $\gcd(f,f')$ divides $f$ which isn't irreducible. Therefore $f'(a)=f(a)=0$ and $f$ irreducible implies $\gcd(f,f')=f$ which implies $f'=0$ ie. $f \in K[x^p]$.

Whence $p$ divides $\deg(f)=[K(a):K]$ which divides $[F:K]$.