I have the following polynomial
$$g(x)=x^6+x^5+x^4+x^2 \in ℤ \bmod 2$$
I'm having trouble finding information on how
to reduce this polynomial, Or let alone being
able to tell if it is reducible in the first place.
If the polynomial has a root then it is reducible correct
So
$$g(0)=0^6+0^5+0^4+0^2=0 \in ℤ \bmod 2$$
$$g(1)=1^6+1^5+1^4+1^2=0 \in ℤ \bmod 2$$
So it is reducible?
EDIT: Okay it's reducible I guess i'm just confused on how to actually reduce it now.
You can begin with factoring out $x^2$: $$x^6+x^5+x^4+x^2=x^2(x^4+x^3+x^2+1).$$ Now observe that $w=1$ is a root of the second factor, so it is divisible by $x-1=x+1$ (we're in characteristic $2$). Indeed, using synthetic division, we readily obtain $$x^6+x^5+x^4+x^2=x^2(x+1)(x^3+x+1),$$ and this last factor is irreducible since it has degree $3$ and no root in $\mathbf F_2$.