# Complete Residue System Modulo $z$ in the Gaussian Integers

Let $$z \neq 0$$ be a non-unit Gaussian integer. I wish to prove that the Gaussian integers in the fundamental parallelogram associated to $$z$$ are a complete residue system modulo $$z$$. I have succeeded in proving that the Gaussian integers inside the fundamental parallellogram are incongruent to each other, by showing that their difference cannot be an integral multiple of $$z$$. Now, I wish to show that for every Gaussian integer, there is a point in the fundamental parallelogram to which it is congruent modulo $$z$$. I have given the following argument, by feel it is "hand-wavey" and wish to make it more rigorous:

Any Gaussian integer $$q$$ can be located inside exactly one of the translates (or on its boundary) of the fundamental parallelogram; with the Division Algorithm, $$q = \beta z + r,$$ where $$\beta z$$ is the closest vertex in the translate and $$r$$ is the vector from this corner to $$q$$. There is a point $$p$$ located in the corresponding position inside the fundamental parallelogram, given by $$p = \beta' z + r,$$ where $$\beta' z$$ is the closest vertex in the fundamental parallelogram and $$r$$ is the same vector from this corner to $$\mathcal{p}.$$ Clearly, then, $$q \equiv_{z} p$$.

How can I formalise this proof?

• Generally, see here on Hermite normal form. Sep 24, 2023 at 2:35