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 p and an integer a, the quadratic Gauss sum g(a; p) is defined as where \zeta_p is a primitive pth root of unity, for example. Equivalently, For a divisible by p, and we have and thus For a not divisible by p, we have, implying that where is the Gauss sum defined for any character χ modulo p.

Properties

p > 2 can be reduced to the case is given by the formula: 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) . G(a,b,c) depends only on the residue class of a and b modulo c . a, b, c, d with gcd(c, d) = 1 one has G(a, b, c) = 0 if gcd(a, c) > 1 except if gcd(a,c) divides b in which case one has gcd(a, c) = 1 . a, b, c be integers with ac ≠ 0 and ac + b even. One has the following analogue of the quadratic reciprocity law for (even more general) Gauss sums m . The values of Gauss sums with and gcd(a, c) = 1 are explicitly given by (a⁄c) is the Jacobi symbol. This is the famous formula of Carl Friedrich Gauss. b > 0 the Gauss sums can easily be computed by completing the square in most cases. This fails however in some cases (for example, c even and b odd), which can be computed relatively easy by other means. For example, if c is odd and gcd(a, c) = 1 one has 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, 2m) = 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 . For every q, the equation an2 + bn + q = 0 has at most two solutions in \mathbb{Z}/2n\mathbb{Z} . Indeed, if n_1 and n_2 are two solutions of same parity, then for some integer \alpha, but is odd, hence. 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, 2m) = 0 . gcd(a, c) = 1 , then k ≥ 2 and an odd prime number p, and for k ≥ 4 and .