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  "body": "\n## Definition\n\nLet $R$ be an [[integral domain|09S3PFBEPM]].\nAn element $p \\in R$ is irreducible if it has these three properties.\n\n1. $p \\neq 0$.\n2. $p$ is not a [[unit|019NFGFPR5]].\n3. If $p = a b$ with $a, b \\in R$, then $a$ is a unit or $b$ is a unit.\n\n## Remarks\n\nThe irreducible elements of $\\mathbb{Z}$ are the numbers $p$ and $-p$ with $p$ prime.\nIn $\\mathbb{Z}[i]$, the number $3$ is irreducible, but $5 = (2 + i)(2 - i)$ is not.\nEvery [[prime element|0MNHXBSX3M]] is irreducible. The converse [[can fail|0CHE3J7QMX]].\nFor polynomials over a field, the irreducible elements are the [[irreducible polynomials|00X2A9G9W3]].\n\n## Explanations\n\n### L1 human\n\nAn irreducible element cannot be split into smaller factors.\nThe only factorizations are trivial ones, with a unit as one factor, like $7 = (-1)(-7)$.\nIn the integers, the irreducible elements are the primes and their negatives.\n",
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