Solution

Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 7 / 4 / Solution

Quadratic periods modulo eleven

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{title2=$G(\chi)$}
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For $\omega=e^{2\pi i/p}$ and the <Legendre symbol> $\chi$, the quadratic Gauss sum is
$$
G(\chi)=\sum_{a=1}^{p-1}\chi(a)\omega^a.
$$
It equals $\sqrt p$ for $p\equiv1\pmod4$ and $i\sqrt p$ for $p\equiv3\pmod4$ with the standard choice of square root.

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