A field of characteristic zero is perfect

Question

How do you prove that a field $F$ of characteristic zero is perfect, or rather that every irreducible $f(x)$ in $F[x]$ is separable?
Thank you!

Answer 1

This is true because every irreducible polynomial $f(x)$ in $F[x]$ is separable (provided the characteristic of $F$ is zero, or $F^p=F$ for prime characteristic $p$). Indeed, we have $f'(x)\neq 0$ for the derivative, because $deg(f')=deg(f)-1$. Here we have used that a polynomial $f(x)$ is inseparable if and only if $f'(x)=0$.