Gaussian Period - General Definition

General Definition

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.