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  "body": "\n## Definition\n\nThe norm on the [[Gaussian integers|0K5W69AC0J]] is the map $N \\colon \\mathbb{Z}[i] \\to \\mathbb{N}$,\n$$N(a + b i) = a^2 + b^2 = (a + b i)(a - b i).$$\n\n## Remarks\n\nThe norm is multiplicative: $N(z w) = N(z) N(w)$ for all $z, w \\in \\mathbb{Z}[i]$. This follows from $|z w| = |z| |w|$ in $\\mathbb{C}$.\n$N(z) = 0$ if and only if $z = 0$.\nThe norm is a Euclidean function on $\\mathbb{Z}[i]$.\nFor example, $N(2 + i) = 5$ and $N(1 + i) = 2$.\n\n## Explanations\n\n### L1 human\n\nThe norm is the squared distance from $0$ to the point $a + b i$ in the complex plane.\nBecause it is multiplicative, it turns questions about factoring Gaussian integers into questions about factoring integers.\n",
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