What are the elements of $\mathbb{F}_3[X]/(X^3-3)$?
A similar question was posted here: Elements of the field $F_2[x] / (x^3 + x + 1)$, but it doesn't explain why the elements of that field look that way.
Anyway the problem I'm trying to solve here is to determine for which $p\in\{2,3,5,7\}$ the rings $\mathbb{F}_p[X]/(X^3-3)$ and $\mathbb{F}_p[T]/(T^3+2)$ are isomorphic.
I don't see how the linked post can help with this question, though.
Hints. For $p=2$, they are not isomorphic. (Look for nilpotents.)
Similarly for $p=3$.
They are equal for $p=5$, and isomorphic for $p=7$ (since both polynomials are irreducible over $\mathbb F_7$.)