For $p$ prime number, let $I=\langle x^3-x\rangle$ and $J=\langle x^3+x\rangle$ be ideals of $\mathbb{Z}_p[x]$ and consider the rings $R=\mathbb{Z}_p[x]/I$ and $S=\mathbb{Z}_p[x]/J$ and the polynomial $f(x)=x^2+x+1 \in \mathbb{Z}_p[x]$.
- Find prime number $p$ so that $f(x)+I$ is invertible in $R$.
- Find prime number $p$ so that $f(x)+J$ is invertible in $S$.
- Find prime number $p>2$ so that the rings $R,S$ are isomorphic.
So far for the first one I showed that for $f(x)+I$ to be invertible in each ring it suffices to show that the greatest common divisor $(x^2+x+1, x^3-x)=1$ and so $p \equiv 1\pmod{3}$. For the second I have that the greatest common divisor $(x^2+x+1, x^3+x)=1$ in any case, using Euclidean algorithm of polynomial division.
For the third question, I should use the first isomorphism theorem of rings, but I have no clue how to use to it here. I think that I should use something from the first two questions, but still no idea.