$\mathbb{Z}[i]$ is Euclidean

Division. Let $z,w\in\mathbb{Z}[i]$, $w\neq0$. Write $z/w=x+yi\in\mathbb{Q}[i]$. Let $q=m+ni$ with $m,n$ the nearest integers to $x,y$. Then $|x-m|\le1/2$, $|y-n|\le1/2$.

Remainder. Let $r=z-qw$. Then $N(r)=N(w)N(z/w-q)=N(w)[(x-m)^2+(y-n)^2]\le N(w)(1/4+1/4)=N(w)/2<N(w)$.

Conclusion. For every $z,w$ with $w\neq0$, there exist $q,r\in\mathbb{Z}[i]$ with $z=qw+r$ and $N(r)<N(w)$. So $\mathbb{Z}[i]$ is Euclidean, hence a PID and UFD.