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).
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
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.
(That could be derived also by ramification arguments in algebraic number theory; see quadratic field.)
As Gauss eventually showed, to evaluate P − P*, the correct square root to take is the positive (resp.
are ubiquitous in number theory; for example they occur significantly in the functional equations of L-functions.
(Gauss sums are in a sense the finite field analogues of the gamma function.
can thus be written as a linear combination of Gaussian periods (with coefficients χ(a)); the converse is also true, as a consequence of the orthogonality relations for the group (Z/nZ)×.
The Gaussian periods generally lie in smaller fields, since for example when n is a prime p, the values χ(a) are (p − 1)-th roots of unity.