Q:

Splitting Field of $x^3-x$ over $\mathbb{F}_4$

What is the splitting field of $p(x)=x^3-x$ over $\mathbb{F}_4$? 
$\mathbb{F}_2=\{0,1\}$ seems also to be the splitting field of $p(x)$
$p(0)=0$
$p(1)=0$
Which one I choose? $\mathbb{F}_4$ or $\mathbb{F}_2$?

A:

The answer has to be an extension of $\mathbb{F}_4$, and therefore it cannot be $\mathbb{F}_2$. But, in $\mathbb{F}_4$,$$x^3-x=x(x^2+1)=x(x+1)^2$$and therefore the answer is $\mathbb{F}_4$.

