Consider $\mathbb{F}_3(\alpha)$ where $\alpha^3 - \alpha +1 = 0$ and $\mathbb{F}_3(\beta)$ where $\beta^3 - \beta^2 +1 =0$.
I know these two fields are isomorphic but I have difficulty buliding an isomorphism between them.
I know I have to determine where $\alpha$ is mapped to under the isomorphism map but I can't figure it out.
Any help is much appreciated.