Prove that $\Bbb F_2(\sqrt[3] x)$ is separable over $\Bbb F_2(x)$.
Now I was trying taking $X^3 -x\in \Bbb F_2( x)[X] $ then if I show this polynomial is separable then I am done. Now $X^3 -x=(X-x^{1/3})(X^2 +aX +b) \Rightarrow b=x^{2/3}, a=x^{1/3}$.
So considering $X^2+x^{1/3}X+x^{2/3}$ we can't factorize using Shreedhar Acharya's rule as $char (F)=2$. So taking $X^2+x^{1/3}X+x^{2/3}=(X-c)(X-d)$ where $c,d \in \Bbb F_2(\sqrt[3] x)$ what will be the solution of $c,d$?