Real cubic subfield of the seventh cyclotomic field
= Real cubic subfield of the seventh cyclotomic field
{title2=$\theta=\zeta_7+\zeta_7^{-1}$}
The unique cubic subfield is the maximal real subfield
$$
\mathbb Q(\theta),
\qquad
\theta=\zeta_7+\zeta_7^{-1},
$$
and
$$
m_{\theta,\mathbb Q}(X)=X^3+X^2-2X-1.
$$
It is the fixed field of complex conjugation.