Prove that the rings $\frac{\Bbb F_3[X]}{(X^3+X^2+2)}$ and $\frac{\Bbb F_3[X]}{(X^3+2X+2)}$ are isomorphic.
I tried to construct an explicit isomorphism using a similar technique as used in this and arrived at the fact that if $\tau$ is one such isomorphism from $\frac{\Bbb F_3[X]}{(X^3+2X+2)}$ to $\frac{\Bbb F_3[X]}{(X^3+X^2+2)}$, then $\tau(\beta)=1+\alpha+\alpha^2$.
But, now I am confused about how to prove that it will indeed generate an isomorphism. I am pretty sure, it's not much, but I can't figure out how.