Gauss sum

The case originally considered by Carl Friedrich Gauss was the quadratic Gauss sum, for the field of residues modulo a prime number , and the Legendre symbol. In this case Gauss proved that or for congruent to 1 or 3 modulo 4 respectively (the quadratic Gauss sum can also be evaluated by Fourier analysis as well as by contour integration). An alternate form for this Gauss sum is :\sum e^{2 \pi i r^2/p}. Quadratic Gauss sums are closely connected with the theory of theta functions. The general theory of Gauss sums was developed in the early 19th century, with the use of Jacobi sums and their prime decomposition in cyclotomic fields. Gauss sums over a residue ring of integers are linear combinations of closely related sums called Gaussian periods. The absolute value of Gauss sums is usually found as an application of Plancherel's theorem on finite groups. In the case where is a field of elements and is nontrivial, the absolute value is . The determination of the exact value of general Gauss sums, following the result of Gauss on the quadratic case, is a long-standing issue. For some cases see Kummer sum. ==Properties of Gauss sums of Dirichlet characters==

The Gauss sum of a Dirichlet character modulo is :G(\chi)=\sum_{a=1}^N\chi(a)e^{2\pi ia/N}. If is also primitive, then :|G(\chi)|=\sqrt{N}, in particular, it is nonzero. More generally, if is the conductor of and is the primitive Dirichlet character modulo that induces , then the Gauss sum of is related to that of by :G(\chi)=\mu\left(\frac{N}{N_0}\right)\chi_0\left(\frac{N}{N_0}\right)G\left(\chi_0\right) where is the Möbius function. Consequently, is non-zero precisely when is squarefree and relatively prime to . Other relations between and Gauss sums of other characters include :G(\overline{\chi})=\chi(-1)\overline{G(\chi)}, where is the complex conjugate Dirichlet character, and if is a Dirichlet character modulo such that and are relatively prime, then : G\left(\chi\chi^\prime\right) = \chi\left(N^\prime\right) \chi^\prime(N) G(\chi) G\left(\chi^\prime\right). The relation among , , and when and are of the same modulus (and is primitive) is measured by the Jacobi sum . Specifically, :G\left(\chi\chi^\prime\right)=\frac{G(\chi)G\left(\chi^\prime\right)}{J\left(\chi,\chi^\prime\right)}. ==Further properties==

• Gauss sums can be used to prove quadratic reciprocity, cubic reciprocity, and quartic reciprocity. • Gauss sums can be used to calculate the number of solutions of polynomial equations over finite fields, and thus can be used to calculate certain zeta functions. ==See also==