Prove that $\mathbb{Z}[i]/(3)$ and $\mathbb{Z}[i]/(7)$ are finite fields and find their cardinality In my rings subject's test I had to prove that $\mathbb{Z}[i]/(21)$ was decomposed as a product of two finite fields, and that was easy to prove for me because $21 = 3\cdot 7$, and $\mathbb{Z}[i]$ is a PID and $3$ and $7$ are irreducible, so by the Chinese Remainder Theorem, and because $(3)$ and $(7)$ are maximal, we have the decomposition into two fields. 
What I am unable to do is to find why they are finite fields and in such case, I had to find their cardinality. Can you help me/give some hints?

There is a short way to see it:
$$\mathbb{Z}[i]/(3) \cong \mathbb{Z}[X]/(X^2+1,3) \cong (\mathbb{Z}[X]/(3)) /(X^2+1) \cong \mathbb{F}_3 / (X^2+1)$$
Where $\mathbb{F}_3$ is the field with three elements.
As one can see easily, this is just $\mathbb{F}_3 [i]$, a field with 9 elements. The other case with 7 works out similarly, I think you can fill in the details that are still needed. Greetings,
Markus Zetto
