Which one of these rings are isomorphic?

Question

We have the follwoing ring and we need to see which one are isomorphic:-

• $\mathbb{Z[i]/(5)}$
• $F_{5}[X]/(X^2-1)$
• $F_{5}[X]/(X^2+1)$

First I thought the first one is field. but later based on answer and comments I knew that all of them are not field even they are not integral domain. So this approach not work.

Answer 1

$p=5$ is not a prime in $\mathbb{Z}[i]$, so the ideal $(5)$ is not maximal. Indeed, $5=(2+i)(2-i)$. Similarly $x^2+1=(x+2)(x+3)$ over $\mathbb{F}_5$. The first and the third are isomorphic, since $\mathbb{Z}[i]\simeq \mathbb{Z}[x]/(x^2+1)$. What about the second one ?