How to show that the splitting field of $x^7 - 5$ over $\mathbb{Q}$ is equal to $\mathbb{Q}(\sqrt[7]{5}, \exp(2\pi i/7))$?

Question

I have a polynomial $x^7 – 5$ over $\mathbb{Q}$. I know the splitting field $K = \mathbb{Q}(\alpha,\alpha\omega, …, \alpha\omega^6)$, where $\alpha = \sqrt[7]{5}$ and $\omega = \exp(2\pi i/7)$.

Now, I know that $K$ is a field but I want to show that $K = \mathbb{Q}\alpha, \omega)$, but now I’m stuck. I was able to show that $\mathbb{Q}(\alpha, \omega) \subseteq \mathbb{Q}(\alpha,\alpha\omega, …, \alpha\omega^6)$ but I don’t know how to show the other inclusion, more specifically, I don’t know how to show that $\omega^2 \in \mathbb{Q}(\alpha, \omega)$. Any help is greatly appreciated.