How can one show that no cyclotomic field contains $\sqrt[3]{2}$?

Question

This is a paraphrase of a problem from an old exam that I'm going over.

Show that for all positive integers $t$, when $\omega$ is a primitive $t$-th root of unity, $\sqrt[3]{2}$ does not lie in the field $\mathbf{Q}\hspace{.02 in}(\omega)$.

How would one show that?

Answer 1

The Galois group of $\mathbb{Q}(\omega)$ over $\mathbb{Q}$ is abelian, so every subextension is also Galois. However, $\mathbb{Q}(\sqrt[3]{2})$ is not Galois over $\mathbb{Q}$.