TITLE: Applications of Gauss sums
QUESTION [28 upvotes]: 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?

REPLY [5 votes]: A small additional note, in line with an earlier answer: Gauss sums are, literally, the Lagrange resolvents obtained in the course of expressing roots of unity in terms of radicals. (Yes, then the Kummer-Stickelberger business can be used to effectively obtain the actual radical expressions...:  here .)