Enumerate the elements of a quotient Gaussian Integers ring

Question

I want to find an extensive list of all the elements of the quotient ring $Z[i]/(3+i)$. Since the Gaussian integers are an euclidean domain with euclidean function $N(a+bi)=a^2+b^2$ the representative of each class is going to be the remainder when we divide by $3+i$, this is, all the elements whose norm is less than $N(3+i)=10$. When we list them all we get that

$Z[i]/(3+i)=\{0,1,-1,i,-i,2,-2,2i,-2i,3,-3,3i,-3i,1+i,1-i,\ldots,2+2i\}$

But this, honestly, looks wrong. Is there any way to get the list of classes in a simple manner that can be generalized to every ideal of $Z[i]$?

Thanks in advance.

Edit: My guess is that since the division algorithm in the gaussian integers does not ensure uniqueness of quotient or remainder some of those classes that I have listed are actually the same class, for example 3=-i, since 3+i=0. But I do not know how to see this in a quick simple manner.

Answer 1

A way to look at $\Bbb{Z}[i]/\langle 3+i\rangle$ is to realize that \begin{align*} 3+i & \equiv 0\\ 3 & \equiv -i\\ 9 & \equiv -1\\ 10 &\equiv 0. \end{align*} So in this quotient ring, we can say $$a+ib \equiv a-3b \pmod{10}.$$ So each element in the quotient ring is equivalent to a remainder class modulo $10$. For example, \begin{align*} 1+2i & \equiv 1-3(2)\equiv 5 \pmod{10}\\ 2+i & \equiv 2-3(1)\equiv 9 \pmod{10}\\ \end{align*} Now see if you can list all the distinct elements.

Note: In fact you can use this idea to get $f(a+ib)=a-3b \pmod{10}$ as a map from $\Bbb{Z}[i]/\langle 3+i\rangle$ to $\Bbb{Z}_{10}$ to show that $\Bbb{Z}[i]/\langle 3+i\rangle \cong \Bbb{Z}_{10}$.

Answer 2

Let $a+bi$ be a nonzero element of $\Bbb Z[i]$. Define $g=\gcd(a,b)$ and $m=(a^2+b^2)/g$. Then $$S=\{u+vi:u,v\in\Bbb Z,0\le u<m,0\le v<g\}$$ is a system of coset representatives for $(a+bi)$ in $\Bbb Z[i]$.

In your example, $g=1$ and $m=10$ and you can take $S=\{0,1,2,3,4,5,6,7,8,9\}$.