Consider the Field $F = \mathbb{Z}_3$ and let $f(x) = x^3 + 2x + 1 \in F[x]$.
We assume known that $K = F(\alpha)$, for $\alpha = x + f(x) \in K$. Use $\alpha$ to write a basis for $K$ over $F$. Express $\alpha^6$ and $\alpha^4$ in this basis.
I get the basis $\left(1, \alpha, \alpha^2\right)$. And then to express $\alpha^4$ and $\alpha^6$ I get:
$$\alpha^4 = \alpha^2 + 1 - \alpha$$ $$\alpha^6 = \alpha^4 + \alpha^2 - \alpha^3$$
Is this correct?