Gaussian period

In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discrete Fourier transform). They are basic in the classical theory called cyclotomy. Closely related is the Gauss sum, a type of exponential sum which is a linear combination of periods. As the name suggests, the periods were introduced by Gauss and were the basis for his theory of compass and straightedge construction. For example, the construction of the heptadecagon (a formula that furthered his reputation) depended on the algebra of such periods, of which is an example involving the seventeenth root of unity Given an integer n > 1, let H be any subgroup of the multiplicative group of invertible residues modulo n, and let A Gaussian period P is a sum of the primitive n-th roots of unity , where runs through all of the elements in a fixed coset of H in G. The definition of P can also be stated in terms of the field trace. We have for some subfield L of Q(ζ) and some j coprime to n. This corresponds to the previous definition by identifying G and H with the Galois groups of Q(ζ)/Q and Q(ζ)/L, respectively. The choice of j determines the choice of coset of H in G in the previous definition. The situation is simplest when n is a prime number p > 2. In that case G is cyclic of order p − 1, and has one subgroup H of order d for every factor d of p − 1. For example, we can take H of index two. In that case H consists of the quadratic residues modulo p. Corresponding to this H we have the Gaussian period summed over (p − 1)/2 quadratic residues, and the other period P* summed over the (p − 1)/2 quadratic non-residues. It is easy to see that since the left-hand side adds all the primitive p-th roots of 1. We also know, from the trace definition, that P lies in a quadratic extension of Q. Therefore, as Gauss knew, P satisfies a quadratic equation with integer coefficients.