Quadratic Gaussian period in the seventh cyclotomic field
= Quadratic Gaussian period in the seventh cyclotomic field
{c}
{title2=$\eta=\zeta_7+\zeta_7^2+\zeta_7^4$}
The unique quadratic subfield of $\mathbb Q(\zeta_7)$ is
$$
\mathbb Q(\eta)=\mathbb Q(\sqrt{-7}),
\qquad
\eta=\frac{-1+\sqrt{-7}}2.
$$
The conjugate period is $-1-\eta$, and
$$
m_{\eta,\mathbb Q}(X)=X^2+X+2.
$$