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

For an odd prime number and an integer, the quadratic Gauss sum is defined as
where is a primitive th root of unity, for example.
Equivalently, we can write this using the Legendre symbol as
For divisible by, and we have and thus
For not divisible by, we have, implying that
where
is the Gauss sum defined for any character modulo.

Properties

  • The value of the Gauss sum is an algebraic integer in the th cyclotomic field.
  • The evaluation of the Gauss sum for an integer not divisible by a prime can be reduced to the case :
  • The exact value of the Gauss sum for is given by the formula:
; Remark
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.

Generalized quadratic Gauss sums

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