<s>[INST] Factoring $X^{16}+X$ over $\mathbb{F}_2$ I just asked wolframalpha to factor $X^{16}+X$ over $\mathbb{F}_2$. The normal factorization is 
$$
X(X+1)(X^2-X+1)(X^4-X^3+X^2-X+1)(X^8+X^7-X^5-X^4-X^3+X+1)
$$
and over $GF(2)$ it is
$$
X(X+1)(X^2+X+1)(X^4+X+1)(X^4+X^3+1)(X^4+X^3+X^2+X+1).
$$
Does the second form follow from the first, or is there a different way to factor over $\mathbb{F}_2$? I noticed that simply replacing the $-$ signs with $+$ signs in the first factorization doesn't yield the second one.
 [/INST] The second form does follow from the first. Note that over $\mathbb F_2$, the pair of polynomials 
$$
X^2 - X + 1 \quad \text{ and } \quad X^2 + X + 1
$$
and the other pair
$$
X^4 - X^3 + X^2 - X + 1 \quad \text{ and } \quad X^4 + X^3 + X^2 + X + 1
$$
are the same. The only difference that comes up in $\mathbb F_2$ is that the polynomial of degree $8$ factors.
Hope that helps,
</s>