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  "body": "\n## Description\n\nThe ring of Gaussian integers is\n$$\\mathbb{Z}[i] = \\{a + b i : a, b \\in \\mathbb{Z}\\},$$\nwhere $i \\in \\mathbb{C}$ satisfies $i^2 = -1$.\nIt is a [[subring|0859D58WPD]] of $\\mathbb{C}$, with $(a + b i)(c + d i) = (a c - b d) + (a d + b c) i$.\nAs a subring of a field, it is an [[integral domain|09S3PFBEPM]].\n\n## Remarks\n\nThe [[norm|03AN1V4ZFJ]] $N(a + b i) = a^2 + b^2$ is multiplicative.\nThe [[units of $\\mathbb{Z}[i]$|06RH4XAGRY]] are $1, -1, i, -i$.\nThe ring $\\mathbb{Z}[i]$ is a [[Euclidean domain|0PN13BB7BT]], so it has unique factorization.\nA prime number $p$ is a sum of two squares if and only if $p$ is not prime in $\\mathbb{Z}[i]$. This holds exactly for $p = 2$ and the primes $p \\equiv 1 \\pmod 4$.\n\n## Explanations\n\n### L1 human\n\nThe Gaussian integers are the points with integer coordinates in the complex plane.\nYou add and multiply them as complex numbers.\nThey form a ring with a theory of primes and factorization, like the integers.\n",
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