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  "body": "\n## Definition\n\nA unique factorisation domain is an [[integral domain|09S3PFBEPM]] $R$ with these two properties.\n\n1. Every nonzero element $a \\in R$ that is not a unit is a product $a = p_1 p_2 \\cdots p_n$ of [[irreducible elements|052SDEPZH7]], with $n \\ge 1$.\n2. The factorization is unique up to order and units. If $p_1 \\cdots p_n = q_1 \\cdots q_m$ with all $p_i, q_j$ irreducible, then $n = m$. Also, there is a permutation $\\sigma$ of $\\{1, \\dots, n\\}$ such that $p_i$ and $q_{\\sigma(i)}$ are [[associates|053E86NWVK]] for each $i$.\n\n## Remarks\n\nThe integers are a unique factorisation domain: this is the fundamental theorem of arithmetic.\nEvery [[principal ideal domain is a unique factorisation domain|055HT85T3S]], and so is every [[polynomial ring over a unique factorisation domain|0D6KH8Y59N]].\nThe ring $\\mathbb{Z}[\\sqrt{-5}]$ is [[not a unique factorisation domain|0BSJYCG7NM]].\nIn a unique factorisation domain, [[irreducible elements are prime|0QRCFZKXH8]].\n\n## Explanations\n\n### L1 human\n\nIn the integers, $60 = 2 \\cdot 2 \\cdot 3 \\cdot 5$, and every other factorization into primes uses the same primes, up to order and signs.\nA unique factorisation domain is a ring where this works for every nonzero nonunit.\nUnits like $-1$ do not count as a different factorization.\n",
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