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 ${\displaystyle \zeta ^{a}}$, where ${\displaystyle a}$ runs through all of the elements in a fixed coset of H in G.

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