Let us consider $x^8 -1$. I want to decompose it over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_4$.
In $\mathbb{F}_3$ I have no problem, since I can use the cyclotomic cosets.
In $\mathbb{F}_2$, I have $$ (\star)\quad \quad (x^8 - 1)= (x^4 - 1) (x^4 +1) = (x^2 - 1)(x^2 + 1)(x^4 +1) = ... = (x-1)^8$$
so the splitting field is $\mathbb{F}_2$ itself and the decomposition is the above one.
- Now, about $\mathbb{F}_4$, I have that $\mathbb{F}_4$ is an extension of $\mathbb{F}_2$ and therefore it contains its splitting field and therefore the decomposition is again $\star$.
Is the points for $\mathbb{F}_3, \mathbb{F}_4$ okay?