How to factor this polynomial over $\mathbb{F}_2$ On page 587 in Dummit and Foote, the authors say the polynomial $\frac{x^{16}-x}{x(x-1)(x^2+x+1)}$ can be factored into quartics over $\mathbb{F}_2$ as $(x^4+x^3+x^2+x+1)(x^4+x^3+1)(x^4+x+1)$.
I am having trouble seeing this. When I divide the polynomial $x^{16}-x$ by $x(x-1)(x^2+x+1)$ using long division, I get $x^{12}+x^9+x^6+x^3+1$. However, I am not sure how to factor this polynomial into quartics. How do I do this? 

The Berlekamp algorithm gives
$$
x^{16}-x=(x^4 + x^3 + x^2 + x + 1)(x^4 + x^3 + 1)(x^4 + x + 1)(x^2 + x + 1)(x + 1)x.
$$
So the statement in Dummit and Foote is correct.
