Prime and maximal ideals of $\mathbb C[\mathbb Z_3]$

Question

I'm trying to find these ideals without success.

I have a hint: $\mathbb C[\mathbb Z_3] \simeq \frac{\mathbb C[X]}{(X^3 - 1)}$, considering $\mathbb C[X]$ is a PID.

Edit: How can I prove that ideals of $\frac{\mathbb C[X]}{(X^3 - 1)}$ are the ideals of $\mathbb C[X]$ containing $(X^3 - 1)$?

Answer 1

In your edit you are just quoting the correspondence theorem which is available everywhere.

In a polynomial ring like $\mathbb C[x]$ that amounts to factoring $x^3-1$ over $\mathbb C$. The maximal ideals will correspond to irreducible factors. You should find exactly three.

Because this ring is Artinian (it's finite dimensional over a field after all) its prime ideals will be exactly its maximal ideals.