Example. [2106E]

Keep the notation in Example 2106 (4). We have $$ \mathrm{Aut}\left( \mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 }) / \mathbb{Q} \right) = \left< \sigma_{1}, \sigma_{2} \right> \cong S_{3}, $$ with $$ \begin{align*} \sigma_1 : && \sqrt[3]{2} &\longmapsto \sqrt[3]{2}\frac{-1 + \sqrt{-3}}{2}, & \sigma_2 : && \sqrt[3]{2} &\longmapsto \sqrt[3]{2}, \\ && \frac{-1 + \sqrt{-3}}{2} &\longmapsto \frac{-1 + \sqrt{-3}}{2}, & && \frac{-1 + \sqrt{-3}}{2} &\longmapsto \frac{-1 - \sqrt{-3}}{2}. \end{align*} $$ One can check that $$ \begin{align*} \mathrm{Aut}\left( \mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 }) / \mathbb{Q}(\sqrt[3]{ 2 }) \right) &= \left< \sigma_{2} \right> \quad (\cong \mathbb{Z} / 2\mathbb{Z}), \\ \mathrm{Aut}\left( \mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 }) / \mathbb{Q}\left( \sqrt[3]{ 2 } \frac{-1 + \sqrt{ -3 }}{2} \right) \right) & = \left< \sigma_{2} \sigma_{1} \right> \quad (\cong \mathbb{Z} / 2\mathbb{Z}), \\ \mathrm{Aut}\left( \mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 }) / \mathbb{Q}\left( \sqrt[3]{ 2 } \frac{-1 - \sqrt{ -3 }}{2} \right) \right) & = \left< \sigma_{2} \sigma_{1}^{2} \right> \quad (\cong \mathbb{Z} / 2\mathbb{Z}), \\ \mathrm{Aut}\left( \mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 }) / \mathbb{Q}(\sqrt{ -3 }) \right) & = \left< \sigma_{1} \right> \quad (\cong \mathbb{Z} / 3\mathbb{Z}), \end{align*} $$ and $$ \begin{align*} \mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 })^{\left< \sigma_{2} \right> } &= \mathbb{Q}(\sqrt[3]{ 2 }), \\ \mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 })^{\left< \sigma_{2} \sigma_{1} \right> } &= \mathbb{Q}\left( \sqrt[3]{ 2 } \frac{-1 + \sqrt{ -3 }}{2} \right), \\ \mathbb{Q}(\sqrt[3]{ 2 }. \sqrt{ -3 })^{\left< \sigma_{2}\sigma_{1}^{2} \right> } &= \mathbb{Q}\left( \sqrt[3]{ 2 } \frac{-1 - \sqrt{ -3 }}{2} \right), \\ \mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 })^{\left< \sigma_{1} \right> } &= \mathbb{Q}(\sqrt{ -3 }), \\ \mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 })^{\left< \sigma_{1}, \sigma_{2} \right> } &= \mathbb{Q}. \end{align*} $$