In number theory, quadratic Gauss sums are certain finite sums of roots of unity. 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 sumg(a; p) is defined as
where is a primitive pth root of unity, for example . Equivalently,
For a divisible by p the expression evaluates to . Hence, we have
For a not divisible by p, this expression reduces to
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.
Let a, b, c be natural numbers. The generalized quadratic Gauss sumG(a, b, c) is defined by
The classical quadratic Gauss sum is the sum g(a, p) = G(a, 0, p).
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