Let $F$ be a splitting field for $x^4 + x^3 + 2x^2 + 1$ over $\mathbb{Z}_3$, find $[F:\mathbb{Z}_3]$.
I'm not sure how to proceed in $\mathbb{Z}_3$, I guess the situation is quite different than with polynomials over $\mathbb{Q}$. Any help is appreciated.