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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.

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. Evaluating the square of the sum P is connected with the problem of counting how many quadratic residues between 1 and p − 1 are succeeded by quadratic residues. The solution is elementary (as we would now say, it computes a local zeta function, for a curve that is a conic). One has

(P − P*)2 = p or −p, for p = 4m + 1 or 4m + 3 respectively. This therefore gives us the precise information about which quadratic field lies in Q(ζ). (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. i times positive real) one, in the two cases. Thus the explicit value of the period P is given by

The Gaussian periods are related to the Gauss sums for which the character χ is trivial on H. Such χ take the same value at all elements a in a fixed coset of H in G. For example, the quadratic character mod p described above takes the value 1 at each quadratic residue, and takes the value 1 at each quadratic non residue. The Gauss sum 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)×. In other words, the Gaussian periods and Gauss sums are each other's Fourier transforms. 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. On the other hand, Gauss sums have nicer algebraic properties.