Exercise 9.2.16

Proof. If $p=7$, and $\zeta =e^{2\mathit{i\pi }∕7}$, the 2-periods corresponding to $H_2=\{-1,1\}=\{1,6\}$ are $(2,1)=\zeta +{\zeta }^{-1},(2,2)={\zeta }^2+{\zeta }^{-2},(2,3)={\zeta }^3+{\zeta }^{-3}$. By Proposition 9.2.6, they are the roots of the irreducible polynomial

$3$ is a primitive root modulo 7. Let $\sigma$ the $\mathbb{Q}$-automorphism determined by $\sigma (\zeta )={\zeta }^3$. Then $\sigma$ gives the chain $(2,1)\mapsto (2,3)\mapsto (2,2)\mapsto (2,1)$, so

By summation of these equalities,

Finally

Therefore $f=x^3+x^2-2x-1$ is the minimal polynomial of $(2,1)=2\cos <!--\; FUNCTION\; APPLICATION\; -->(2\pi ∕7)$ over $\mathbb{Q}$ (and also of $(2,2),(2,3)$).

The fixed field $L_2$ of ${\tilde{H}}_2$ corresponding to $H_2$ is $\mathbb{Q}(\zeta +{\zeta }^{-1})$, of degree 3 over $\mathbb{Q}$, and ${\tilde{H}}_2=\{e,\tau \}$, where $\tau (\zeta )={\zeta }^{-1}=\bar{\zeta }$, so $\tau$ is the restriction of the complex conjugation to $L_2$. The end of the proof is the same as in Exercise 3. □