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  "body": "\n## Statement\n\nLet $K$ be a [[field|04C5ZBG1WT]] and let $a, b, c \\in K[t]$ be polynomials in the [[polynomial ring|0SDC934R52]] with $a + b = c$.\nAssume that $a$, $b$, $c$ are pairwise coprime and that not all of the [[formal derivatives|0EGN1E1JPB]] $a'$, $b'$, $c'$ are zero.\nFor a nonzero polynomial $f$, let $\\operatorname{rad}(f)$ be the product of the distinct monic [[irreducible|03FKX7Z5DC]] factors of $f$.\nThen\n$$\\max\\set{\\deg a, \\deg b, \\deg c} \\le \\deg \\operatorname{rad}(abc) - 1.$$\n\n## Remarks\n\nWhen $K$ is algebraically closed, $\\deg \\operatorname{rad}(abc)$ is the number of distinct roots of $abc$.\nIf $K$ has characteristic $0$, the hypothesis on derivatives just says that $a$, $b$, $c$ are not all constant.\nA short proof, due to Noah Snyder, studies the Wronskian $ab' - a'b$: it is nonzero, it is divisible by $abc/\\operatorname{rad}(abc)$, and its degree is less than $\\deg a + \\deg b$.\nExample: over $\\Q$, take $a = t^2$, $b = 2t + 1$ and $c = (t + 1)^2$; then $\\operatorname{rad}(abc) = t(2t + 1)(t + 1)$ has degree $3$, and the bound $2 \\le 3 - 1$ is sharp.\nThe derivative hypothesis is necessary: over a field of characteristic $p \u003e 0$, take $a = t^p$, $b = 1$, $c = (t + 1)^p$; then $a' = b' = c' = 0$, the radical has degree $2$, and $\\deg c = p \u003e 1$.\nA consequence: for $n \\ge 3$ there are no pairwise coprime polynomials $x, y, z$ over $\\C$, not all constant, with $x^n + y^n = z^n$.\n\n## Explanations\n\n### L1 human\n\nIf two polynomials add up to a third, and they share no common factor, then the three together must have many different roots.\nMore precisely, the number of distinct roots is larger than every one of the degrees.\nSo you cannot make all three polynomials into high powers of a few simple factors.\n\n### L2 human\n\nThe theorem is the polynomial form of the [[abc conjecture|G7JH2NXPC0]], with the degree playing the role of the logarithm of the size and the radical counting distinct prime factors.\nIn contrast with the integer case, it has a short and complete proof by differentiation.\nIt gives quick proofs of Fermat's Last Theorem for polynomials and of related results on equations such as $x^2 - y^3 = $ a polynomial.\n",
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