Let $F=\mathbb Z/(2)$. The splitting field of $x^3+x^2+1\in F[x]$ is a finite field with eight elements.
my attempt of solution:
If $\alpha$ is a root in this polynomial in its splitting field, then I would like to prove that $F(\alpha)$ is the splitting field.
what I get is $x^3+x^2+1=(x-\alpha)(x^2+(1+\alpha)x+(\alpha +\alpha^2))$.
I'm trying to find the root of $x^2+(1+\alpha)x+(\alpha +\alpha^2)$, maybe it's a multiple of $\alpha$.
I need help!