Gaussian Period

A Gaussian period is a sum of roots of unity whose exponents range over a coset of a subgroup of the multiplicative group of residue classes. For an odd prime , write with positive integers and , choose a primitive root modulo , and let be a primitive root of unity. The periods are

(1)

where , 1, ..., . Each contains terms, and together their exponents partition the nonzero residue classes modulo , so

(2)

The automorphism of the cyclotomic field cyclically permutes the periods. They are Galois conjugates, and each generates an extension field of degree over .

For , , and , the periods sum the quadratic residues and quadratic nonresidues, respectively. They satisfy and , giving

			(3)
			(4)

Successively splitting periods into two smaller periods produces quadratic equations over the fields already constructed. For a Fermat prime, repeated splitting ends with two-term periods . This gives the quadratic extensions used in constructing the regular heptadecagon and 257-gon.