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  "body": "\n## Statement\n\nThe ring of [[Gaussian integers|0K5W69AC0J]] $\\mathbb{Z}[i]$ is a [[Euclidean domain|0GJ2HRG95T]], with the [[norm|03AN1V4ZFJ]] $N(a + b i) = a^2 + b^2$ as Euclidean function.\nThat is, for all $z, w \\in \\mathbb{Z}[i]$ with $w \\neq 0$, there are $q, r \\in \\mathbb{Z}[i]$ with $z = q w + r$ and $N(r) \u003c N(w)$.\n\n## Remarks\n\nThe proof: write $z / w = x + y i$ in $\\mathbb{C}$. Let $q = m + n i$, where $m$ and $n$ are integers nearest to $x$ and $y$.\nThen $|z/w - q|^2 \\le \\tfrac{1}{4} + \\tfrac{1}{4} = \\tfrac{1}{2}$, so $r = z - q w$ has $N(r) \\le \\tfrac{1}{2} N(w) \u003c N(w)$.\nA consequence is that $\\mathbb{Z}[i]$ is a principal ideal domain and a unique factorization domain.\n\n## Explanations\n\n### L1 human\n\nTo divide $z$ by $w$, compute the complex quotient and round both coordinates to the nearest integer.\nThe rounding error is small, so the remainder is smaller than $w$ in norm.\nThis makes the Euclidean algorithm work for Gaussian integers.\n",
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