Euclidean algorithm in lattices in $\mathbb{C}$ Let $R=\mathbb{Z}[i]$ be the ring of Gaussian integers. I want to prove that, for every $\alpha,\beta\in R,\beta\neq 0$, there exist $\gamma,\delta\in R$ such that $\alpha=\gamma\beta+\delta$, with $|\delta|<|\beta|$, where
$$|a+ib|:=a^2+b^2$$
for every $a+ib\in R$.
To do this, for every $\alpha,\beta\in R,\beta\neq 0$ I defined a function
$$f_{\alpha,\beta}(\gamma):=\alpha-\gamma\beta$$
for every $\gamma\in R$. Then $f\equiv f_{\alpha,\beta}$ is the composition of a dilation by the factor $\sqrt{|\beta|}$, a rotation and a translation.
My guess is that there exists $\gamma\in R$ such that $|f(\gamma)|$ is strictly smaller than $|\beta|$. If so, i can use this $\gamma$ and its image $f(\gamma)$ in the division algorithm equation. Do such a $\gamma$ exist?
 A: One way to find such $\gamma, \delta$ is to simply construct them, having a good idea of what you want them to be like beforehand. We want $\gamma$ to essentially be the quotient $\alpha/\beta$ but it can't (usually) be that perfectly since we also require $\gamma\in R.$ So we just do the best we can and choose $\delta$ to fix things up:
Let $\alpha = a+bi, \beta = c+di.$ Then $$ \frac{\alpha}{\beta} = \frac{ac+bd}{c^2+d^2} + i \frac{bc-ad}{c^2+d^2}.$$
Let $\gamma=x+iy$ where $x$ is the nearest integer to $\dfrac{ac+bd}{c^2+d^2}$ and $y$ is the nearest integer to $\dfrac{bc-ad}{c^2+d^2}.$
Then $\alpha/\beta = \gamma + p+qi$ where $|p|,|q|\leq 1/2.$
So $\alpha = \beta \gamma + \delta$ where $\delta = cp-dq +i(qc+dp)$. We have $\delta\in R$ since $\delta = \alpha - \beta \gamma$ so we just need to verify that $|\delta|<|\beta|$ i.e that $(cp-dq)^2+(qc+dp)^2 < c^2+d^2.$ This is true since the left hand side expands to $$c^2p^2-2cdpq+d^2q^2+c^2q^2+2cdpq+d^2p^2 = (c^2+d^2)(p^2+q^2)\leq \frac{c^2+d^2}{2}.$$
