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  "body": "\n## Summary\n\nA Jacobi sum is a sum $\\sum_{a} \\chi(a)\\lambda(1 - a)$ over a [[finite field|0ZXZ82FTX2]], built from two multiplicative characters $\\chi$ and $\\lambda$.\nIt counts, in a weighted way, the solutions of $x + y = 1$.\nWhen $\\chi$, $\\lambda$ and $\\chi\\lambda$ are nontrivial, it is a quotient of [[Gauss sums|EQ6A3ZN48K]].\nThen its absolute value is the square root of the field size.\nJacobi sums are used to count points on curves such as $x^n + y^n = 1$ over finite fields and to prove [[cubic|EMVECB93EQ]] and [[quartic reciprocity|W0M20JWC9K]].\n\n## Explanations\n\n### L1 human\n\nGauss sums involve complex roots of unity of order $p$, which can be awkward.\nJacobi sums use only the values of multiplicative characters, so they live in a smaller ring of roots of unity.\nFor the Legendre symbol modulo $5$, the Jacobi sum is $\\sum_a \\left(\\frac{a}{5}\\right)\\left(\\frac{1-a}{5}\\right) = -1$.\nWriting a prime as a sum of two squares, or as $x^2 + 3y^2$, can be done by computing a suitable Jacobi sum.\n",
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