Subgroup

Fixed Subfield

$G=\langle \sigma \eta \rangle$

$\mathbb{Q}$

$\langle \sigma {\eta }^2\rangle$

$\mathbb{Q}\left(\cos \frac{2\pi }{5}\right)=\mathbb{Q}\left(\sqrt{5}\right)$

$\langle \sigma \rangle$

$\mathbb{Q}\left(i\sin \frac{2\pi }{5}\right)=\mathbb{Q}\left(\omega \right)$

$\langle \eta \rangle$

$\mathbb{Q}\left(\cos \frac{2\pi }{11}\right)=\mathbb{Q}\left(a\right)$

$\langle {\eta }^2\rangle$

$\mathbb{Q}\left(\cos \frac{2\pi }{11},\cos \frac{2\pi }{5}\right)=\mathbb{Q}\left(a,\sqrt{5}\right)$

$\left\{e\right\}$

$\mathbb{Q}\left(\cos \frac{2\pi }{11},\omega \right)$