Splitting Field and Irreducibility

Question

Let $f(x) = x^3 − x + 1 ∈ F_3[x]$. I wish to show that $[L : F_3] = 3$ where $L$ is the splitting field of $f(x)$ over $F_3$. First, I can show $f(x)$ is irreducible over $F_3$ right since it has no roots there. Then, I am not sure~

Answer 1

If $\alpha$ is a root, then $\alpha+1$ and $\alpha-1$ are roots, too (easy to check); so $L=K[\alpha]$. Hence, $[L:\mathbb F_3]=3$.