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  "body": "\n## Summary\n\nThe greatest common divisor of two polynomials over a field is the monic common divisor that every common divisor divides.\nIt is the [[greatest common divisor|0ZAF9KHDDF]] in the ring $K[x]$, made unique by asking it to be monic.\nThe [[Euclidean algorithm|023VY493K0]] computes it with repeated [[long division|D6D0SR1FK3]].\nIt detects common roots, and $\\gcd(f, f')$ detects [[repeated roots|067F8YVYMH]].\n\n## Explanations\n\n### L1 human\n\nTwo polynomials may share factors, as two integers may share prime factors.\nThe greatest common divisor is the product of all the shared factors, counted as often as both contain them.\nFor example, $x^2 - 1 = (x - 1)(x + 1)$ and $x^2 - 3x + 2 = (x - 1)(x - 2)$ share only $x - 1$, which is their greatest common divisor.\nThe [[definition|99J1WXG3F7]] gives the conditions.\n",
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