Why did people constructed Quadratic Gauss Sum?

Question

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.

Answer 1

One application of Gauss sums is finding intermediate fields between $\mathbf{Q}$ and $\mathbf{Q}(\zeta_n)$.

Here is a link for more information.