Seventh cyclotomic field ($\mathbb{Q}({\zeta }_7)$)

For a primitive seventh root of unity ${\zeta }_7$,where ${\sigma }_a({\zeta }_7)={\zeta }_7^a$. Its four intermediate fields correspond to the four subgroups of the cyclic group $C_6$.

The unique quadratic subfield of $\mathbb{Q}({\zeta }_7)$ isThe conjugate period is $-1-\eta$, and

The unique cubic subfield is the maximal real subfieldandIt is the fixed field of complex conjugation.