How do you factorize a polynom in $\mathbb{Z}_2$? How can you efficiently factorize a polynom (in $\mathbb{Z}_2$) ?
Example in this answer:
$$x^{16}-x=x (x + 1) (x^2 + x + 1) (x^4 + x + 1) (x^4 + x^3 + 1) (x^4 + x^3 + x^2 + x + 1)$$
How do you do this by hand?

Partial answer:
We know that $x-1 =x+1$, so we have:
\begin{eqnarray}
x^{16}-x &=& x(x^{15}-1)\\
&=& x(x^5-1)\underbrace{(x^{10}+x^5+1)}_{p(x)}\\
&=& x(x-1)(x^4+x^3+x^2+x+1)p(x)\\
\end{eqnarray}
Now we have to factor somehow $p(x)$...
