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I was studying number theory these days where Quadratic Gauss sum came up. https://en.wikipedia.org/wiki/Quadratic_Gauss_sum

My question was that:

  1. What motivated them to construct Gauss Sum in the first place.

  2. Why did they use $(\frac{t}{p})l^{at}$ rather than simply say $(\frac{t}{p})l^{t}$ in the first place.

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    The Gauss sum is a special case of the Gauss sum of a Dirichlet character, which appears naturally in functional equations for L-functions. I don't know if this is where they were born. You may want to ask this at https://hsm.stackexchange.com/ – Bart Michels Apr 17 '18 at 17:13
  • A simple proof of the quadratic reciprocity theorem.
    • – Jack D'Aurizio Apr 17 '18 at 17:40
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  • The book by Ireland and Rosen on number theory has applications of Gauss sums almost everywhere.
    • – Dietrich Burde Apr 17 '18 at 18:20
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    @barto@DietrichBurde Thanks! The short answer was in the web barto given https://hsm.stackexchange.com/questions/2133/what-motivated-gauss-quadratic-sums and https://math.stackexchange.com/questions/11675/applications-of-gauss-sums –  Apr 17 '18 at 18:38