Gaussian period

With

ζ = e^{2 πi /7}

, put

α = ζ + ζ^{2} + ζ^{4}

and

β = ζ^{3} + ζ^{5} + ζ^{6}

β = \overline{α}

α + β = - 1

, and

α^{2} = α + 2 β

. Consequently

α^{2} + α + 2 = 0

. The positive imaginary part of

α

fixes

α = (- 1 + i 7) /2

and

β = (- 1 - i 7) /2

. They are the nonidentity kernel values of the two degree-three irreducible characters of the nonabelian group of order 21.

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A Gaussian period is a mathematical concept that arises in number theory, specifically in the study of algebraic integers within cyclotomic fields. In particular, a Gaussian period is associated with the Gaussian integers, which are complex numbers of the form \( a + bi \), where \( a \) and \( b \) are integers and \( i \) is the imaginary unit.