One explanation is given in this math.SE answer.In the language of that answer, you want to describe the unique quadratic subfield of $\mathbb{Q}(\zeta_p)$.

Oct 27, 2016 · How to calculate Gauss sum? Ask Question Asked 8 years, 5 months ago. Modified 8 years, 3 months ago.

Nov 24, 2010 · The totally mysterious explanation is that Gauss sums naturally appear when you start thinking about the discrete Fourier transform. For example, the trace of the DFT matrix is a Gauss sum. But more mysteriously, Gauss sums are eigenfunctions of the DFT in a certain sense. (I sketch how this works here.)

May 11, 2016 · Writing a Gauss sum as a sum over divisors. 2. An identity involving Gauss sums and convolution. 0

From here you can use reduction properties of the quadratic Gauss sum and the Chinese Remainder Theorem to prove the even cases. There is no general formula for a generalized Gauss sum. You can find a reciprocity theorem for these sums in the book Gauss and Jacobi Sums as well, also in Introduction to Analytic Number Theory by Apostol.

The wikipedia page for Quadratic Gauss Sum says that the fact that $g(a;p)^2 = \left(\frac{-1}{p}\right)p$ is simple, yet I am having trouble coming up with a proof.

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Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Feb 7, 2020 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

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