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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]], and let $m, n \\ge 0$ be integers with $m + n \\ge 1$.\nLet $f = a_m x^m + \\dots + a_0$ and $g = b_n x^n + \\dots + b_0$ be elements of $R[x]$.\nThe Sylvester matrix $S(f, g)$ is the $(m + n) \\times (m + n)$ matrix whose rows are:\n\n1. for $i = 1, \\dots, n$, the row $(a_m, a_{m-1}, \\dots, a_0)$ preceded by $i - 1$ zeros and filled up with zeros on the right;\n2. for $j = 1, \\dots, m$, the row $(b_n, b_{n-1}, \\dots, b_0)$ preceded by $j - 1$ zeros and filled up with zeros on the right.\n\nThe resultant of $f$ and $g$ is $\\operatorname{Res}(f, g) = \\det S(f, g)$, the [[determinant|0C3CFHDN6R]] of this matrix.\n\n## Remarks\n\nExample: for $f = x - a$ and $g = x - b$, the matrix is $\\begin{pmatrix} 1 \u0026 -a \\\\ 1 \u0026 -b \\end{pmatrix}$, and $\\operatorname{Res}(f, g) = a - b$.\nLet $R = K$ be a field with $a_m \\ne 0$ and $b_n \\ne 0$, and let $\\alpha_1, \\dots, \\alpha_m$ and $\\beta_1, \\dots, \\beta_n$ be the roots of $f$ and $g$ in a [[splitting field|02NZZ1J0FH]] of $f g$.\nThen\n$$\\operatorname{Res}(f, g) = a_m^n \\, b_n^m \\prod_{i=1}^{m} \\prod_{j=1}^{n} (\\alpha_i - \\beta_j).$$\nSo $\\operatorname{Res}(f, g) = 0$ if and only if $f$ and $g$ have a common root, if and only if they have a common factor of positive degree.\nFor instance, $\\operatorname{Res}(x^2 - 1, x - 1) = 0$, because of the common root $1$.\nThe value depends on the chosen formal degrees $m$ and $n$: if $a_m = b_n = 0$, the first column is zero and the resultant is $0$ even without a common root.\nFor $f$ of degree $n$ with leading coefficient $a_n$, $\\operatorname{Res}(f, f') = (-1)^{n(n-1)/2} a_n \\operatorname{Disc}(f)$, where $\\operatorname{Disc}$ is the [[discriminant|0CCPN09EMJ]].\n\n## Explanations\n\n### L1 human\n\nThe resultant is the determinant of a square table filled with shifted copies of the two coefficient lists.\nIt is zero exactly when the two polynomials share a root, as long as both leading coefficients are nonzero.\n",
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