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?

Start with the factorization of  $x^{15}-1$ into cyclotomic polynomials:
$$
x^{16}-x = x(x^{15}-1)=x \Phi_1(x) \Phi_3(x) \Phi_5(x) \Phi_{15}(x) 
$$
which gives
$$
x^{16}-x = x(x - 1)(x^2 + x + 1) (x^4 + x^3 + x^2 + x + 1) (x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)
$$
We then need to further factor these mod 2. The last factor is the only one that is reducible. Guessing a factorization into two quartics works.
