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 sum g(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


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.

Generalized quadratic Gauss sums

Let a, b, c be natural numbers. The generalized quadratic Gauss sum G(a, b, c) is defined by


The classical quadratic Gauss sum is the sum g(a, p) = G(a, 0, p).

This is a direct consequence of the Chinese remainder theorem.
Thus in the evaluation of quadratic Gauss sums one may always assume gcd(a, c) = 1.
for every odd integer m. The values of Gauss sums with b = 0 and gcd(a, c) = 1 are explicitly given by
Here (a/c) is the Jacobi symbol. This is the famous formula of Carl Friedrich Gauss.
where ψ(a) is some number with 4ψ(a)a ≡ 1 (mod c). As another example, if 4 divides c and b is odd and as always gcd(a, c) = 1 then G(a, b, c) = 0. This can, for example, be proved as follows: because of the multiplicative property of Gauss sums we only have to show that G(a, b, 2n) = 0 if n > 1 and a, b are odd with gcd(a, c) = 1. If b is odd then an2 + bn is even for all 0 ≤ n < c − 1. By Hensel's lemma, for every q, the equation an2 + bn + q = 0 has at most two solutions in /2n.[clarification needed] Because of a counting argument an2 + bn runs through all even residue classes modulo c exactly two times. The geometric sum formula then shows that G(a, b, 2n) = 0.
If c is not squarefree then the right side vanishes while the left side does not. Often the right sum is also called a quadratic Gauss sum.
holds for k ≥ 2 and an odd prime number p, and for k ≥ 4 and p = 2.

See also


  1. ^ M. Murty, S. Pathak, The Mathematics Student Vol. 86, Nos. 1-2, January-June (2017), xx-yy ISSN: 0025-5742
  2. ^ Theorem 1.2.2 in B. C. Berndt, R. J. Evans, K. S. Williams, Gauss and Jacobi Sums, John Wiley and Sons, (1998).