Let $F= \Bbb Z_2[x]/ \langle f \rangle$ with $f=x^3+x+1 \in \Bbb Z_2[x]$. Now consider f as an element of $F[x]$ and
a) show that there exists $\alpha \in F$ with $f(a)=0$
b) find $g \in F[x]$ with $f=(x-\alpha)g$
c) show that also $\alpha^2$ and $\alpha^4$ are roots of $f$ over $F$ and write $f$ as a product of irreducible elements of $F[x]$
So I tried finding the root of $f$ over $F$ and it seems like $x^2+x$ and $x^2$ are both roots but I'm stuck on part b and c, when I divide $f$ by $x-\alpha$ I always get a remainder.
Am I approaching this the right way or maybe I'm missing something. Also I'm having some trouble wrapping my head around the $F[x]$, what should be the modulus of it?
Thanks in advance!