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  "body": "\n## Statement\n\nLet $R$ be a [[principal ideal domain|0RZYDVZ475]].\nThen $R$ is a [[unique factorisation domain|06YMXJMXYS]].\n\n## Remarks\n\nExistence of factorisations: a principal ideal domain is Noetherian, so a process of splitting an element into factors must stop.\nUniqueness: in a principal ideal domain, an irreducible element generates a maximal ideal, so it is prime.\nConsequences: $\\mathbb{Z}$, $\\mathbb{Z}[i]$ and $K[x]$ for a field $K$ have unique factorisation.\nThe converse is false: $\\mathbb{Z}[x]$ is a unique factorisation domain but [[not a principal ideal domain|06E3AG9TM2]].\n\n## Explanations\n\n### L1 human\n\nIn a principal ideal domain, every irreducible element behaves like a prime number.\nThat is the key to unique factorisation.\nSo every Euclidean domain, such as the integers, has unique factorisation.\n",
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