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  "body": "\n## Statement\n\nLet $R$ be a [[Euclidean domain|0GJ2HRG95T]] and let $a, b \\in R$.\nThen $a$ and $b$ have a [[greatest common divisor|0F7KNJ1ZYN]] $d$, and there are $s, t \\in R$ with\n$$d = s a + t b.$$\n\n## Remarks\n\nThe [[Euclidean algorithm|023VY493K0]] computes $d$. The extended Euclidean algorithm also computes $s$ and $t$, by back substitution.\nIn $\\mathbb{Z}$: $\\gcd(252, 198) = 18 = 4 \\cdot 252 - 5 \\cdot 198$.\nThe same identity holds in every [[principal ideal domain|0RZYDVZ475]], because the ideal $(a, b)$ is principal.\nIf $d \\neq 0$, the coefficients are not unique: $s + k b/d$ and $t - k a/d$ also work for each $k \\in R$.\n\n## Explanations\n\n### L1 human\n\nThe gcd of two elements is a combination of them with coefficients in the ring.\nFor example, $\\gcd(4, 6) = 2 = 2 \\cdot 4 - 1 \\cdot 6$.\nThis is the main tool to solve linear equations $s a + t b = c$ in a Euclidean domain.\n",
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  "msc": [
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  "name": "Bézout's identity in a Euclidean domain",
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