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  "body": "\n## Definition\n\nA Euclidean domain is an [[integral domain|09S3PFBEPM]] $R$ with a function $\\delta \\colon R \\setminus \\{0\\} \\to \\mathbb{N}$ with this property.\nFor all $a, b \\in R$ with $b \\neq 0$, there are $q, r \\in R$ such that $a = b q + r$, and $r = 0$ or $\\delta(r) \u003c \\delta(b)$.\nThe function $\\delta$ is a Euclidean function of $R$.\n\n## Remarks\n\nThe absolute value is a Euclidean function on $\\mathbb{Z}$. The degree is a Euclidean function on $K[x]$ for a field $K$.\nSome authors also require $\\delta(a) \\le \\delta(a b)$ for all nonzero $a, b$. This changes the class of Euclidean domains only in appearance, because a ring with a Euclidean function has one with this extra property.\nEvery Euclidean domain is a [[principal ideal domain|0RZYDVZ475]].\n\n## Explanations\n\n### L1 human\n\nIn a Euclidean domain, you can always divide with a remainder that is smaller than the divisor.\nThe function $\\delta$ measures what smaller means.\nThis is all you need to run the Euclidean algorithm.\n",
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