I need to decide if the splitting field of the polynomial $x^6 + x^3 + 2$ over $\mathbb{F}_3$ is isomorphic to $\mathbb{F}_{3^2}$.
My argument is that $\mathbb{F}_{3^2}$ is splitting field of the polynomial $x^{3^2}-x$ over $\mathbb{F}_{3}$, and since $x^6 + x^3 + 2 \nmid x^{3^2}-x$ then I say that the splitting field of $x^6 + x^3 + 2$ is not $\mathbb{F}_{3^2}$. Is this correct? Maybe there is a way to prove this question by extracting the splitting field of $x^6 + x^3 + 2$ over $\mathbb{F}_3$ by hand but I can't figure it out.
Thanks!