Prove: 2^(1/3) cannot be written in terms of any given root of unity

Question

How does one prove that there exists no natural number $n$ such that $\sqrt[3]{2} $ belongs to the field extension of the rationals by the $n$th root of unity?

Answer 1

Consider the Euler totient function and use Galois theory.

Answer 2

If $\zeta_n$ is a primitive $n$-th root of unity, the extension $\mathbb Q(\zeta_n)\supset\mathbb Q$ is abelian. If $\root3\of2$ were in one such cyclotomic field, so would its splitting field, which is $\mathbb Q(\zeta_3,\root3\of2)$. But this field is nonabelian over $\mathbb Q$. No go.