Are the quotient rings $\mathbb{Z}_2[x]/\left<x^2\right>$ and $\mathbb{Z}_2[x]/\left<x^2 + 1\right>$ isomorphic?
Attempt. I think they are isomorphic.
I have thought it extensively. I have supposed a isomorphic function $\phi$ as follows.
$\phi (x + 1) \to x$
$\phi (x) \to (x + 1)$
$\phi (1) \to 1$
$\phi (0) \to 0$
Then I verified in brute force manner that the mapping is homomorphic.
Can anyone please tell me if I have gone wrong anywhere?
Your brute force approach is fine. Alternatively, note that the map $$\Bbb{Z}_2[x]\ \longmapsto\ \Bbb{Z}_2[x]:\ x\ \longmapsto\ x+1,$$ is a ring isomorphism. The preimage of the ideal $\langle x^2+1\rangle$ is then $$\langle (x-1)^2+1\rangle=\langle x^2\rangle,$$ from which it follows that the corresponding quotients are isomorphic.