Let $K$ be the splitting field of $f(X)=X^3-2$ over $\mathbb{Q}$. I am asked to find complete list of intermediate fields $k$, $\mathbb{Q}\subseteq k\subseteq K$ such that $[k:\mathbb{Q}]=3$.
I've showed that $K=\mathbb{Q}(\sqrt[3]{2},i\sqrt{3})$ and that the Galois group of $K/\mathbb{Q}$ is isomorphic to $D_6(\cong S_3)$ with the following generators
$$\rho:\sqrt[3]{2}\mapsto \sqrt[3]{2}\omega, \quad i\sqrt{3}\mapsto i\sqrt{3}$$ $$\sigma:\sqrt[3]{2}\mapsto \sqrt[3]{2}, \quad i\sqrt{3}\mapsto -i\sqrt{3},$$
where $\omega$ is the 3rd root of unity $e^{2\pi i/3}$. Now by the fundamental theorem of Galois theory, the only such intermediate field is given by the fixed field of $\sigma$, which is $\mathbb{Q}(\sqrt[3]{2})$. However, don't the fields $\mathbb{Q}(\sqrt[3]{2}\omega)$ and $\mathbb{Q}(\sqrt[3]{2}\omega^2)$ also satify the desired properties? What exactly am I doing wrong? Thanks in advance...