A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general Gauss sum.
These objects are named after Carl Friedrich Gauss, who studied them extensively and applied them to quadratic, cubic, and biquadratic reciprocity laws.
For an odd prime number p and an integer a, the quadratic Gauss sum g(a; p) is defined as where
is a primitive pth root of unity, for example
, implying that where is the Gauss sum defined for any character χ modulo p. In fact, the identity was easy to prove and led to one of Gauss's proofs of quadratic reciprocity.
However, the determination of the sign of the Gauss sum turned out to be considerably more difficult: Gauss could only establish it after several years' work.
Later, Dirichlet, Kronecker, Schur and other mathematicians found different proofs.