Let $K= \mathbb{Q}[\sqrt{d}]$ where $d$ is a square free negative integer that is either $2$ or $3$ $\mod 4$. (So $\mathcal{O}_K= \mathbb{Z}[\sqrt{d}]$). I proved a theorem as below:
Let $\mathfrak{a} \subseteq \mathcal{O}_K$ an integral ideal. Then there exist $a,b,c \in \mathbb{Z}^{>0}$ such that $\mathfrak{a}=a\mathbb{Z} +(b+c\sqrt{d})\mathbb{Z}$.
In this setup, $a$ is the smallest possible positive integer in the ideal and similarly $c$ is the smallest positive integer such that $b+c\sqrt{d} $ is in the ideal.
Now, I have a problem with the lemma below:
Let we have $\mathfrak{a}=a\mathbb{Z} +(b+c\sqrt{d})\mathbb{Z}$ and consider $\mathcal{O}_K/\mathfrak{a}$. Then coset representatives are given as $\{x+y\sqrt{d}: 0 \le x < a, 0\le y < c \}$ .
I do not understand that part. For example, maybe we have some elements in this set so that they belong the same ideal class. How do we guarantee that these are the all representatives?
