finite fields - Factorization of $x^8-x$ over $F_2$ and $F_4$

How can I factorize $x^8-x$ over the fields $F_2$ and $F_4$?

Answer

Hint: We have that $x^8-x$ splits as
$$ x(x+1)(x^3+x^2+1)(x^3+x+1)$$
over $\mathbb{F}_2$ and $\gcd(x^8-x,x^4-x)=x^2+x=x(x+1)$.