Quadratic Gauss sum


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.

Definition

Let be an odd prime number and an integer. Then the Gauss sum modulo,, is the following sum of the th roots of unity:
If is not divisible by, an alternative expression for the Gauss sum is
Here is the Legendre symbol, which is a quadratic character modulo. An analogous formula with a general character in place of the Legendre symbol defines the Gauss sum.

Properties

Let be natural numbers. The generalized Gauss sum is defined by
The classical Gauss sum is the sum.

Properties