Which of the following quotient rings is not a field?

Question

\begin{align*} Z_3[x]/(x^2+2)\\Z_5[x]/(x^2+3)\\Z_7[x]/(x^2+5)\\Q[x]/(x^2+2)\\R[x]/(x^2+2) \end{align*}

Here $Q$ is the rational numbers and $R$ is the real numbers. Which of the above ones is not a field? I cannot find a way to judge this... Acutually this was the last question of the GRE subject math test I took today. I just picked $Z_5$ randomly...

Answer 1

In $\mathbb{Z}_3[x]$, $x^2+2=x^2-1=(x+1)(x-1)$. So…

Answer 2

In $\Bbb Z_7[x]$, $x^2+5=x^2-9=(x-3)(x+3)$....

Answer 3

What the previous answers want to say (and don't wanna ruin it), is that for a field $k$, the polynomial ring $k[x]$ is well-known to be PID. Therefore any maximal ideal is prime and vice versa. In particular can be proven that an ideal $I \subset k[x]$ is prime, if-f the polynomial $f \in k[x]$ such that $I=<f>$ is irreducible. Now along with that, plus the answers above you have all that you need to decide what is what..