Suppose $\alpha = \omega + \omega^7 + \omega^{11}$, where $\omega$ is a primitive root of unity of order 19. Determine the Galois group of $\mathbb{Q}(\alpha) / \mathbb{Q}$. With which other well known group is $\text{Gal}(\mathbb{Q}(\alpha) / \mathbb{Q})$ isomorphic?
My own work: Since $\mathbb{Q}(\alpha) \subset \mathbb{Q}(\omega)$. It holds that $19=[\mathbb{Q}(\omega) : \mathbb{Q}] = [\mathbb{Q}(\omega) : \mathbb{Q}(\alpha)] \cdot [\mathbb{Q}(\alpha) : \mathbb{Q}]$. Since 19 is a prime number and $ [\mathbb{Q}(\alpha) : \mathbb{Q}] \neq 1$ it holds that $[\mathbb{Q}(\alpha) : \mathbb{Q}]=19$. The only group of order 19 is $\mathbb{Z}/19\mathbb{Z}$.
Is this a correct answer? Is there a way to solve this question using less theory? Thanks!