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  "body": "\n## Summary\n\nA Lagrange resolvent is a sum $\\alpha + \\zeta \\sigma(\\alpha) + \\cdots + \\zeta^{n-1} \\sigma^{n-1}(\\alpha)$ built from an element $\\alpha$, an automorphism $\\sigma$ of order $n$ and an $n$-th [[root of unity|0YV2CKP2AV]] $\\zeta$.\nThe automorphism $\\sigma$ multiplies it by $\\zeta^{-1}$, so its $n$-th power is fixed by $\\sigma$.\nThis turns a [[cyclic extension|VFE5B71E7Y]] into an extension obtained by adjoining an $n$-th root, which is the key step in solving equations by radicals.\nLagrange used such sums of the roots to explain the formulas for cubic and quartic equations.\n\n## Explanations\n\n### L1 human\n\nSuppose the roots of an equation are permuted in a cycle by a symmetry.\nMix the roots with weights $1, \\zeta, \\zeta^2, \\ldots$ that are powers of a root of unity.\nTurning the cycle once only multiplies this mix by a root of unity, so a suitable power of the mix does not change at all.\nThat power is then a known quantity, and the mix is one of its $n$-th roots.\nFor a quadratic with roots $r_1, r_2$ the resolvent is $r_1 - r_2$, and its square $(r_1 - r_2)^2$ is the discriminant, which is why the quadratic formula has a square root.\n",
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