For proving the quadratic reciprocity, Gauss sums are very useful. However this seems an ad-hoc construction. Is this useful in a wider context? What are some other uses for Gauss sums?
Applications of Gauss sums
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number-theory
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1[Related HSM.SE question](http://hsm.stackexchange.com/questions/2133/what-motivated-gauss-quadratic-sums) – 2015-05-10