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  "body": "\n## Summary\n\nA Gauss sum is a finite sum $\\sum_a \\chi(a)\\psi(a)$ over a [[finite field|0ZXZ82FTX2]], where $\\chi$ is a multiplicative character and $\\psi$ an additive character.\nIt mixes the multiplicative and the additive structure of the field.\nIts absolute value is the square root of the field size when both characters are nontrivial.\nGauss sums are a basic tool in proofs of reciprocity laws and in the study of [[character sums|VTYJ9419C7]] and [[Dirichlet L-functions|9D2Z543EJQ]].\n\n## Explanations\n\n### L1 human\n\nTake an odd prime $p$ and the complex number $\\zeta = e^{2\\pi i/p}$, a [[root of unity|0YV2CKP2AV]].\nWeight each power $\\zeta^a$, for $a = 1, \\dots, p - 1$, by $+1$ or $-1$, according to whether $a$ is a square modulo $p$ or not, and add.\nFor $p = 3$ this gives $\\zeta - \\zeta^2 = i\\sqrt{3}$.\nThe surprise is that the sum always has absolute value exactly $\\sqrt{p}$, much smaller than the $p - 1$ terms suggest.\nThis square root is where a lot of number theory hides.\n",
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