Let $p(x) = x^3 + x + 1 \in \mathbb{Z}_2[x]$ and $E = \mathbb{Z}_2[x]/p(x)$. Factor $p(x)$ into linear factors in $E[x]$.
Note that $p(t) = 0$, where $t = x + ⟨p(x)⟩$. You might also wish to show that $p(t^2) = 0$.
So far, I have that $E = \{a+bt+ct^2 : a,b,c \in \mathbb{Z}_2, t^3 = 1+t\}$ and I have shown that $p(t^2) = 0$. But I really do not know how to proceed from here.
I have in my notes that the solution is $p = [x + t][x + t^2][x + (t + t^2)] \in E[x]$ but I do not have any details on how this solution was obtained.
Any help would be appreciated.