The quadratic Gauss sum is defined by
a) Show that
whenever mod\: n)" alt="r\equiv s\mod\: n)" />
and deduce that , where the summation extends over any complete set of residues.
b) Let and be integers with , and let and run through complete sets of residues and .
Prove that runs through a complete set of residues , and that
c) Use the results of parts (a) and (b) to prove that
, for ,
and evaluate each term of this equation when and .