positive characteristic and multiple roots I can't understand a proof in  Milne, proposition 2.12 at pag 29. In particular, i can't prove the implication $c)\Rightarrow d)$ where:
c) $F$ has characteristic $p\neq 0$ and $f$ is a polynomial in $X^p$ ;
d) all the roots of $f$ are multiple
Suppose $f(X)=g(X^p)$ and $g(X)=\displaystyle\prod_i(X-a_i)^{m_i}$ in some extension $K$ of $F$. Then
$$f(X)=g(X^p)=\displaystyle\prod_i(X^p-a_i)^{m_i}=\displaystyle\prod_i(X-\alpha_i)^{pm_i}$$
where $\alpha_i^{p}=a_i$
Well, my question is: who assures me that such an $\alpha_i$ exists? This is the case when $F$ is finite, so that $F=F^p$, but i don't have this hypothesis! 

If they aren't already in $K$ (as when $F$ hence$ K$ are finite), the $\alpha_i$ are in some extension $L$ or $K$.
For example, pick $F = \Bbb F_p(X^p)$. The polynomial $Y^p - X^p$ in $F[Y]$ has $p$ repeated roots in the extension $L = \Bbb F_p(X)$ of $F$ since $Y^p - X^p = (Y-X)^p$
