= Solution
The <Gaussian integers> form an <integral domain>, since they are a <subring> of the <complex numbers>. Use the <Gaussian integer norm>
$$
N(a+bi)=a^2+b^2=|a+bi|^2,
$$
which is a positive <integer> for nonzero $a+bi$ and satisfies $N(\alpha\beta)=N(\alpha)N(\beta)$.
For $\alpha,\beta\in\mathbb Z[i]$ with $\beta\ne0$, write $\alpha/\beta=x+iy$. Choose <integers> $m,n$ with $|x-m|\leq1/2$ and $|y-n|\leq1/2$, put $q=m+in$, and set $r=\alpha-\beta q$. Then
$$
\alpha=\beta q+r,\qquad
N(r)=N(\beta)\big((x-m)^2+(y-n)^2\big)\leq\tfrac12N(\beta)<N(\beta).
$$
Thus division with remainder has either $r=0$ or a strictly smaller <Gaussian integer norm>. This proves that $N$ is a <euclidean function> and
$$
\boxed{\mathbb Z[i]\text{ is a Euclidean domain}}.
$$
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