I'm working on the following homework question:
Let $F = \mathbb{Q}(i, \sqrt{3}, \omega)$ where $i \in \mathbb{C}$ such that $i^2 = -1$ and $\omega$ is a complex (nonreal) cube root of 1. Compute [F:$\mathbb{Q}$] and a basis for $F$ over $\mathbb{Q}$.
I'm just beginning to study Galois theory, so I'm very unfamiliar with how to start this type of problem, so any help would be appreciated.
You should be able to work from the fact that we can choose $$\omega=e^{2i\pi/3}=-1/2+i\frac{\sqrt3}2$$ This is clearly contained in $\mathbb Q(i, \sqrt3) $, so you just have to find a basis of this. Can you prove that $1,i,\sqrt3,i\sqrt3$ form such a basis? The goal is to get all possible elements uniquely. You need to prove that you can get all elements of the field as a rational linear combination of these and no rational linear combination with nonzero coefficients yields zero.