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  "body": "\n## Statement\n\nLet $R$ be a [[unique factorisation domain|06YMXJMXYS]].\nThen the [[polynomial ring|0Q9EZS90GY]] $R[x]$ is a unique factorisation domain.\n\n## Remarks\n\nThe proof uses [[Gauss's lemma|0QFV5Y1Z5X]] and unique factorisation in $K[x]$, where $K$ is the field of fractions of $R$.\nThe irreducible elements of $R[x]$ are the irreducible elements of $R$ and the primitive polynomials of positive degree that are irreducible in $K[x]$.\nBy induction, $R[x_1, \\dots, x_n]$ is a unique factorisation domain. In particular, $\\mathbb{Z}[x]$ and $K[x, y]$ are.\nThese rings are not [[principal ideal domains|0RZYDVZ475]]: $(2, x)$ in $\\mathbb{Z}[x]$ and $(x, y)$ in $K[x, y]$ are not principal.\n\n## Explanations\n\n### L1 human\n\nInteger polynomials factor uniquely into irreducible pieces, like integers do.\nThe pieces are prime numbers and irreducible primitive polynomials.\nFor example, $2x^2 - 2 = 2 (x - 1)(x + 1)$.\n",
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