Here is the question I want to solve:
Show that $\mathbb{Z}[\sqrt{3}]$ satisfies the following division property:
Given $a,b \in \mathbb{Z}[\sqrt{3}]$ there exist $q,r \in \mathbb{Z}[\sqrt{3}]$ such that $b = aq + r$ and $0 \leq |N(r)| \leq |N(a)|.$
Hint: In $\mathbb{Q}[\sqrt{3}]$ we have $s = b/a = s_{1} + s_{2} \sqrt{3}$ for $s_{1},s_{2} \in \mathbb{Q}.$ Write $s_{i} = m_{i} + t_{i}$ for $m_{i} \in \mathbb{Z}$ and $0 \leq |t_{i}| \leq \frac{1}{2}.$ Define $m = m_{1} + m_{2}\sqrt{3}$ and $t = t_{1} + t_{2}\sqrt{3}.$ Then $b = am + at.$ Show that $at \in \mathbb{Z}[\sqrt{3}]$ and $|N(at)|< |N(a)|.$
Define the norm function $N:\mathbb{Q}[\sqrt{3}] \rightarrow \mathbb{Q}$ by $N(a_{1} + a_{2}\sqrt{3}) = \operatorname{det} \phi (a_{1} + a_{2}\sqrt{3}) = a_{1}^2 - 3 a_{2}^2.$ Show that $N(xy) = N(x)N(y)$ for all $x,y \in \mathbb{Q}[\sqrt{3}].$\
My questions are:
1-Is there a typo in the inequality in the definition of the division property? because in the hint it is a strict inequality.
2- I do not understand the importance of the inequality in the definition, could anyone explain that to me please?
2-I do not understand the idea of the hint at all, what are we doing, could anyone explain it to me please? I do not even know how to apply the hint.
3- Is there a simpler way to prove the division property for $\mathbb{Z}[\sqrt{3}]$? If so can anyone show it for me please?
Is this the same as asking me to prove that $\mathbb{Z}[\sqrt{3}]$ is a Euclidean domain?If so what is the importance of expressing the remainder in terms of the norm function?
I found this question here $\mathbb{Z}[\sqrt{11}]$ is norm-euclidean is the answer of mine should be very similar to that?if so I am not understanding the algorithm used, could anyone help me in that please?