Simple exercise on Galois theory Find the splitting field of $x^6-2x^4-8x^2+16$ over $\mathbb {F}_3$ and list the intermediate fields between the base camp and the splitting field.
Hint: Over $\mathbb{F}_3$ we have
$$
x^6-2x^4-8x^2+16=(x^2 + 2x + 2)(x^2 + x + 2)(x^2 + 1).
$$
My factorization was incorrect, I have understood my error. Now to calculate the Galois group of the extension (which is actually a Galois extension, as $\mathbb {F}_9$ is the splitting field of a separable polynomial, having first derivative different from zero, with coefficients in $\mathbb {F}_3$) I just note that the degree of the extension is $[\mathbb {F}_9:\mathbb {F}_3]=2$, so the group has 2 elements and it is isomorphic to $\mathbb {Z}_2$. Is it right?