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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $f = a_n x^n + \\dots + a_0 \\in K[x]$ with $n \\ge 1$ and $a_n \\neq 0$.\nLet $f = a_n (x - r_1) \\cdots (x - r_n)$ in a [[splitting field|02NZZ1J0FH]] of $f$.\nThe discriminant of $f$ is\n$$\\operatorname{Disc}(f) = a_n^{2n-2} \\prod_{1 \\le i \u003c j \\le n} (r_i - r_j)^2.$$\n\n## Remarks\n\nThe product is symmetric in the roots, so $\\operatorname{Disc}(f) \\in K$, by the [[fundamental theorem of symmetric polynomials|0GW1ZZH0R4]]. It is a polynomial with integer coefficients in $a_0, \\dots, a_n$.\nSo $\\operatorname{Disc}(f) = 0$ if and only if $f$ has a [[repeated root|067F8YVYMH]].\nFor $f = a x^2 + b x + c$, $\\operatorname{Disc}(f) = b^2 - 4ac$.\nFor $f = x^3 + p x + q$, $\\operatorname{Disc}(f) = -4p^3 - 27q^2$.\n\n## Explanations\n\n### L1 human\n\nThe discriminant measures how far apart the roots of a polynomial are.\nIt is zero when two roots coincide.\nFor a real quadratic, its sign tells whether the roots are real or complex.\n",
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