How to see $2^{{1}/{3}}$ is not included to $\mathbb Q(w)$?

Question

put, $w=e^{i2\pi/9}$. $\mathbb Q(w)$ is a splitting field of $x^9-1$ on $\mathbb Q$. Then how to see $2^{1/3}$ is not included to $\mathbb Q(w)$?

Answer 1

The Galois closure of $\Bbb Q(2^{1/3})/\Bbb Q$ is a non-Abelian group. The extension $\Bbb Q(w)/\Bbb Q$ has an Abelian Galois group. If $\Bbb Q(2^{1/3})\subseteq\Bbb Q(w)/\Bbb Q$ then the Galois closure of $\Bbb Q(2^{1/3})$ would be contained in $\Bbb Q(w)$. But a subextension of an Abelian extension is Abelian.