In $\mathbb{Q}[x]/(x^3-2)$, we could have $\theta = \sqrt[3]{2}$, but can we not also have $\theta' = \sqrt[3]{2}\omega_3$, where $\omega_3$ is the primitive third root of unity, or $\theta'' = \sqrt[3]{2}\omega_3^2$?
I have learned that the fields $\mathbb{Q}(\sqrt[3]{2})$ and $\mathbb{Q}(\sqrt[3]{2}\omega_3)$ are isomorphic, but not equal. Is there a way to know which field we are talking about? Or is it for some reason irrelevant?