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  "body": "\n## Definition\n\nA Gaussian prime is a prime element of the ring $\\mathbb{Z}[i]$ of [[Gaussian integers|0B268GVX95]].\nThat is, it is a nonzero Gaussian integer $\\pi$ that is not a unit, with the following property.\nFor all Gaussian integers $\\alpha$ and $\\beta$, if $\\pi \\mid \\alpha \\beta$, then $\\pi \\mid \\alpha$ or $\\pi \\mid \\beta$.\n\n## Remarks\n\nSince $\\mathbb{Z}[i]$ is a [[unique factorization domain|0SPYEFH7H4]], the Gaussian primes are also the irreducible elements: the non-units that are not a product of two non-units.\nA Gaussian integer $\\alpha$ is a Gaussian prime if and only if one of two conditions holds.\nEither $N(\\alpha)$ is a prime number, or $\\alpha$ is a unit times a prime $p \\equiv 3 \\pmod 4$.\nFor example, $1 + i$, $2 + i$ and $3$ are Gaussian primes. But $5 = (2 + i)(2 - i)$ is not.\n\n## Explanations\n\n### L1 human\n\nGaussian primes play the role of prime numbers in the Gaussian integers.\nSome ordinary primes, such as $5$, split into two Gaussian primes. Others, such as $3$, stay prime.\n",
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