I'm trying to find how many equivalence classes on the Gaussian integers can be formed just by adding integers (as part of a wider consideration on how many there are altogether).
Let $\gamma \in \mathbb{Z}[i]$
Consider $\mathbb{Z}[i]_{mod\gamma}$.
The set is not infinite so there exists $z \in \mathbb{Z}$ such that $\gamma + z$ loops back to the equivalence class of $[0]_\gamma$. I'm trying to find the lowest such $z$.
So I considered $\gamma = a + bi \Rightarrow \gamma + z = a + z + bi$.
If $\gamma + z \in [0]_\gamma$ then $\frac{(a+z+bi)(a-bi)}{(a+bi)(a-bi)} = \frac{(a^2 + b^2 + za) -zbi}{a^2+b^2} \in \mathbb{Z}[i]$
Since I'm only investigating adding integers for now I considered:
$\frac{a^2 + b^2 + za}{a^2 + b^2} \in \mathbb{Z}$
This implies $a^2 + b^2 + za = (a^2+b^2)k$ for some $k \in \mathbb{Z}$
Rearranging: $za = (a^2+b^2)k - a^2 + b^2 = (a^2+b^2)(k-1)$.
So $za = (a^2+b^2)l$ for some $l \in \mathbb{Z}$
So $z = (a^2 + b^2)\frac{l}{a}$. But this is only true if l is a multiple of a.
So it follows $z = (a^2+b^2)m$ for some $m \in \mathbb{Z}$.
In other words, the integers which wrap back around to the equivalence class of $[0]_\gamma$ when added to $\gamma$ are the multiples of the norm $N(\gamma)$.
Therefore, there should be precisely $N(\gamma)$ equivalence classes that can be made just by adding integers.
The problem is I know (because it was told to me) that the number of equivalence classes is $N(\gamma)$, which means I must have gone wrong somewhere. My method implies that there are more than $N(\gamma)$ equivalence classes since I haven't even begun considering adding Gaussian integers.
Where did I go wrong?
Show that if $\gcd(a,b)=1$ then $(a+ib)\Bbb{Z}[i]\cap \Bbb{Z} = (a^2+b^2)\Bbb{Z}$: if $(a+ib)(c+id)= k$ then $c-id=\frac{c^2+d^2}{k}(a+ib) $, $\frac{c^2+d^2}{k}$ must be an integer from which $c+id$ is a multiple of $a-ib$ and $k$ is a multiple of $a^2+b^2$.
From there $\Bbb{Z}[i]/(a+ib)$ is of characteristic $g ((a/g)^2+(b/g)^2)$ where $g=\gcd(a,b)$.