I would like to find the splitting field of the polynomial $f=(X^3-2)(X^2-2)\in\mathbb{F}_3[X]$ and then calculate its degree over $\mathbb{F}_3$.
I can rewrite this polynomial into $(X^3-2)(X^2-2)=(X^3+1)(X^2-2)=(X+1)^3(X^2-2)$ (since we can apply to Frobenius for $p=3$). Then it seems that $f$ is not irreducible and even already has a root in $2\in\mathbb{F}_3$! Now what comes up in my mind are:
1) The field $\mathbb{F}_3[X]/\langle(X^2-2)\rangle$
2) The field $\mathbb{F}_3[\sqrt2]$
but somehow I feel that both of them are not the correct answer. Once I know what the splitting field looks like then maybe I will have a better idea on what its degree is.
I've already seen a couple of questions which seems relevant, but still can't find a general idea to solve this kind of question. Any hints would be appreciated!