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  "body": "\n## Statement\n\nThe [[units|019NFGFPR5]] of the [[Gaussian integers|0K5W69AC0J]] $\\mathbb{Z}[i]$ are $1$, $-1$, $i$ and $-i$.\n\n## Remarks\n\nThe proof uses the [[norm|03AN1V4ZFJ]]: if $u v = 1$, then $N(u) N(v) = 1$, so $N(u) = 1$. The only solutions of $a^2 + b^2 = 1$ in integers are $(\\pm 1, 0)$ and $(0, \\pm 1)$.\nConversely, an element of norm $1$ is a unit, because $(a + b i)(a - b i) = 1$.\nThe group of units is cyclic of order $4$, generated by $i$.\n\n## Explanations\n\n### L1 human\n\nA unit must have norm $1$, so it lies on the unit circle.\nThe only points with integer coordinates on the unit circle are $1$, $i$, $-1$ and $-i$.\n",
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