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  "body": "\n## Statement\n\nLet $K$ be a [[field|0QA8CHH8MG]].\nThen the polynomial ring $K[x]$ is a [[Euclidean domain|0GJ2HRG95T]], with the [[degree|0DPF6WRZW6]] as Euclidean function.\nThat is, for all $f, g \\in K[x]$ with $g \\neq 0$, there are $q, r \\in K[x]$ with $f = q g + r$ and $r = 0$ or $\\deg r \u003c \\deg g$.\n\n## Remarks\n\nThe ring $K[x]$ is an integral domain because $K$ is one. The existence of $q$ and $r$ is the [[division algorithm|0MES3KS0ZC]] over a field.\nA consequence is that $K[x]$ is a principal ideal domain and a unique factorization domain.\nThe ring $\\mathbb{Z}[x]$ is not a Euclidean domain, because it is [[not a principal ideal domain|06E3AG9TM2]].\n\n## Explanations\n\n### L1 human\n\nOver a field, you can divide any polynomial by any nonzero polynomial and get a remainder of smaller degree.\nThis is the same situation as division with remainder for integers.\nSo all the theory of the Euclidean algorithm works for polynomials over a field.\n",
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