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  "body": "\n## Definition\n\nLet $F$ be a [[finite field|0ZXZ82FTX2]] with $q$ elements.\nA *multiplicative character* of $F$ is a [[group homomorphism|0JYTG5K9Y4]] $\\chi \\colon F^\\times \\to \\C^\\times$; it is extended to $F$ by $\\chi(0) = 0$.\nAn *additive character* of $F$ is a group homomorphism $\\psi \\colon (F, +) \\to \\C^\\times$.\nThe *Gauss sum* of $\\chi$ and $\\psi$ is\n$$g(\\chi, \\psi) = \\sum_{a \\in F} \\chi(a) \\psi(a).$$\n\n## Remarks\n\nThe most common case is $F = \\Z/p\\Z$ for a prime $p$ and $\\psi(a) = \\zeta^a$ with $\\zeta = e^{2\\pi i/p}$; one then writes $g(\\chi) = \\sum_{a=1}^{p-1} \\chi(a) \\zeta^a$.\nIf $\\chi$ and $\\psi$ are both nontrivial, then $\\abs{g(\\chi, \\psi)} = \\sqrt{q}$.\n\nExample: let $p = 3$ and let $\\chi$ be the [[Legendre symbol|0JTC045NNZ]] modulo $3$, so $\\chi(1) = 1$ and $\\chi(2) = -1$.\nThen $g(\\chi) = \\zeta - \\zeta^2 = i\\sqrt{3}$, and $g(\\chi)^2 = -3$, in line with the general rule $g(\\chi)^2 = \\left(\\frac{-1}{p}\\right) p$ for the quadratic character.\n\nNon-example: the sum $\\sum_{a \\in \\Z/p\\Z} \\chi(a) e^{2\\pi i a^2/p}$ is not a Gauss sum in this sense, because $a \\mapsto e^{2\\pi i a^2/p}$ is not additive, so it is not an additive character.\n\nFor a [[Dirichlet character|7XB5VJKDPG]] $\\chi$ modulo $n$ one defines in the same way $g(\\chi) = \\sum_{a \\bmod n} \\chi(a) e^{2\\pi i a/n}$.\n",
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