Quadratic Gauss sum

In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general Gauss sum. These objects are named after Carl Friedrich Gauss, who studied them extensively and applied them to quadratic, cubic, and biquadratic reciprocity laws.

Let p be an odd prime number and a an integer. Then the Gauss sum mod p, g(a;p), is the following sum of the pth roots of unity:

If a is not divisible by p, an alternative expression for the Gauss sum (with the same value and can be done by evaluating $\sum _{n=0}^{p-1}\left(1+\left({\frac {n}{p}}\right)\right)\zeta _{p}^{n}$ $\sum _{n=0}^{p-1}\left(1+\left({\frac {n}{p}}\right)\right)\zeta _{p}^{n}$ in two different ways) is

Here $\chi (n)=\left({\frac {n}{p}}\right)$ $\chi (n)=\left({\frac {n}{p}}\right)$ is the Legendre symbol, which is a quadratic character mod p. An analogous formula with a general character χ in place of the Legendre symbol defines the Gauss sum G(χ).

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