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?

You can view $\mathbb{Z}[i]$ as a 2D grid of points, quotient by the ideal $(3)$ makes it wrap around at 3 so you the $x$ and $y$ coordinates are limited to the set $0,1,2$. In total you have 9 points.
To make this rigorous show that you can take any $x + iy \in \mathbb Z[i]$ and then express it as $x + iy + (3) = x' + i y' + (3)$ with $x', y' \in \{0,1,2\}$. And that each of these $9$ values are distinct.
Note that in $\mathbb{Z}[i]$ the ideal $(3)$ contains values like $3i$.
