Let $E/F$ be a finite extension. TFAE:
(1) $E/F$ is separable.
(2) $E=F(\alpha_1, \cdots, \alpha_n)$ where $\alpha_1, \cdots, \alpha_n$ are distinct and separable over $F$
To show (2) $\Rightarrow$ (1), WLOG assume $ch(F)=p$ for some prime $p$.
We can inductively define $F=E_0, E_i=F(\alpha_1, \cdots, \alpha_i), E_n=E$. A given hint is to show $E_{i+1}=E_i(E_{i+1}^p)$. From the hint I conclude that
$E_{i+1}=E_i(E_{i+1}^p)= \cdots=F(E_1^p)(E_2^p)\cdots(E_{i+1}^p)=F((E_1E_2\cdots E_{i+1})^p)=F(E_{i+1}^p)=F(F(\alpha_1, \cdots, a_{i+1})^p)=F(\alpha_1^p, \cdots, \alpha_{i+1}^p)$
I guess $E_{i+1}^p=E_{i+1}$ so that $E_{i+1}$ is perfect, but I couldn't really show this so far.
I would appreiciate any help but I prefer not to use field embedding for proof cause it wasn't discussed in my class.
Thank you.