# Why did people constructed Quadratic Gauss Sum? [duplicate]

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.

## marked as duplicate by Dietrich Burde number-theory StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Apr 17 '18 at 18:20

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