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This question already has an answer here:

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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marked as duplicate by Dietrich Burde number-theory Apr 17 '18 at 18:20

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ 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 hsm.stackexchange.com $\endgroup$ – punctured dusk Apr 17 '18 at 17:13
  • $\begingroup$ 1. A simple proof of the quadratic reciprocity theorem. $\endgroup$ – Jack D'Aurizio Apr 17 '18 at 17:40
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    $\begingroup$ 2. The book by Ireland and Rosen on number theory has applications of Gauss sums almost everywhere. $\endgroup$ – Dietrich Burde Apr 17 '18 at 18:20
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    $\begingroup$ @barto@DietrichBurde Thanks! The short answer was in the web barto given hsm.stackexchange.com/questions/2133/… and math.stackexchange.com/questions/11675/… $\endgroup$ – user416486 Apr 17 '18 at 18:38
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One application of Gauss sums is finding intermediate fields between $\mathbf{Q}$ and $\mathbf{Q}(\zeta_n)$.

Here is a link for more information.

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