You're on the right track. If is not separable, then write . If is separable then we are done. If not, we can repeat the process, writing as some . Eventually you have to get a separable polynomial, and the result follows immediately.
Let E/F an algebraic field extension and let p=CharF>0 .
Let a be an elemnt in E.
prove that there exists i>=0 such as a^p^i is separable.
i thought to use to fact that if F is a field with Char p >0, and
f(x)=g(x^p) for some g over F, then f(x) (which its root is a) isn't separable..
but i'm not quite sure how to use it...
Thanks in advance