Prove by counterexample that $\gamma$ and $\delta$ are not necessarily unique

Question

Assume $\mathbb Q[\sqrt{d}]$ is a Euclidean Field and $\alpha$, $\beta$ are two quadratic integers in $\mathbb Q[\sqrt{d}]$, so that there exists integers $\gamma$ and $\delta$ in $\mathbb Q[\sqrt{d}]$ so that $\alpha = \gamma \beta + \delta$, and $|N(\delta)|<|N(\beta)|$. How can we prove by counterexample that $\gamma$ and $\delta$ are not necessarily unique?

Answer 1

Let $d=-1$, so that our field is $\mathbb Q[i]$ and the ring of integers is $\mathbb Z[i]$. Then for fixed $\beta$, the numbers $\gamma\beta$ with $\gamma\in\mathbb Z[i]$ form a square lattice. The trick is now to translate this lattice in such a manner that it has two distinct points of minimal norm. So for example we can let $\beta=2$ and translate it by $1+i$ (so that $0$ now is in the middle of a square "mesh" of the lattice). Then the translated lattice points $\pm1\pm i\in(1+i)+2\mathbb Z[i]$ all have norm $N(\pm 1\pm i)=1^2+1^2=2<4=N(2)$. Thus with arbitray $\alpha \in (1+i)+2\mathbb Z[i]$ we will obtain a counterexample as desired, for example

$$\alpha=1+i,\ \beta=2, \ \begin{cases}\gamma=0,&\delta=1+i\\\gamma=1,&\delta=-1+i\\\gamma=i,&\delta=1-i\\\gamma=1+i,&\delta=-1-i\end{cases} $$ This even gives us a "four-fold" counterexample.