= Gaussian periods for seventh roots of unity
{c}
{title2=$\alpha,\beta=(-1\pm i\sqrt7)/2$}
With $\zeta=e^{2\pi i/7}$, put $\alpha=\zeta+\zeta^2+\zeta^4$ and $\beta=\zeta^3+\zeta^5+\zeta^6$. These <Gaussian periods> obey $\beta=\overline\alpha$, $\alpha+\beta=-1$, and $\alpha^2=\alpha+2\beta$. Consequently $\alpha^2+\alpha+2=0$. The positive imaginary part of $\alpha$ fixes $\alpha=(-1+i\sqrt7)/2$ and $\beta=(-1-i\sqrt7)/2$. They are the nonidentity kernel values of the two degree-three <irreducible characters> of the <nonabelian group of order 21>.
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