I have this definition
A field $F$ is perfect if it has characteristic $0$ or if it has characteristic $p$ and $F^p = F$
From Wikipedia, I have this fact about separable polynomials:
Irreducible polynomials over perfect fields are separable.
I can show that if $p(x)$ is irreducible and $char(F) = 0$, then $p$ cannot have multiple zeros, i.e. it's separable. I can also show that, if $p(x)$ has multiple zeros, it has to be of the form $p(x) = q(x^p)$.
Now my question:
In the comment on Wikipedia, how does the second part follow? That is, that if $char(F)=p$ for $p$ prime and $f(x)$ is irreducible over $F$ then it cannot have multiple zeros?
Many thanks for your help.