Adjoining a root of $p$ to $F_p$ is etale?! I'm confused about etale extensions of $F_p$. We know the etale extensions of a field are the products of separable finite field extensions. But if you take $F_p$ and adjoin a root of p this is separable and finite since $F_p$ is perfect. But this extension can't be etale since it is certainly badly ramified.

$p$ is $0$, and if you take $\mathbb{F}_p$ and adjoin a root of $0$ you just get $\mathbb{F}_p$ again.