When considering the Galois group of the splitting field of the polynomial $x^3-2$, it is mentioned in my notes that $\sqrt[3]{2}$ can be mapped to $\sqrt[3]{2}$,$\sqrt[3]{2}\omega$ or $\sqrt[3]{2}\omega^2$, where $\omega$ is the cube root of unity. $\omega$ must be mapped to $\omega$ or $\omega^2$.
My question is why is this so? Sorry for the beginner question, but why can't $\sqrt[3]{2}$ be mapped to say $\omega$, or $\omega$ be mapped to say, 1?
Thank you very much for help.