Been reviewing some Galois theory, and computing Galois groups has been relatively routine, but one has me stumped.
Suppose $\omega$ is a primitive 37th root of unity. Let's set $\alpha=\omega+\omega^{10}+\omega^{26}$. I want to compute the Galois group of $\mathbf{Q}(\alpha)/\mathbf{Q}$.
I realize that $\mathbf{Q}(\alpha)\subseteq\mathbf{Q}(\omega)$, so $[\mathbf{Q}(\alpha):\mathbf{Q}]$ must divide $[\mathbf{Q}(\omega):\mathbf{Q}]=\varphi(37)=36$. So the order of the Galois group must be a divisor of 36, but that's about all I've managed.
What's the trick to finding this Galois group? Thanks!
Edit: Any automorphism has to map $\omega$ to $\omega^k$ where $(k,37)=1$. I noticed that the maps sending $\omega\mapsto\omega^{10}$ and $\omega\mapsto\omega^{26}$ both fix $\alpha$. Where does one go from there?