Roots of Unity in fields Which roots of unity are contained in the fields: $\mathbb{Q}[i]$, $\mathbb{Q}[\sqrt2]$, $\mathbb{Q}[\sqrt3]$, $\mathbb{Q}[\sqrt5]$, $\mathbb{Q}[\sqrt{-2}]$ and $\mathbb{Q}[\sqrt{-3}]$? 
I know that the roots of unity in $\mathbb{Q}[i]$ are $1$, $-1$, $i$, and $-i$. I'm having a hard time finding the roots of unity in the other fields. If anyone could offer any advice, it would be greatly appreciated.

The first three are easy enough so I will tackle the last two. Recall that the degree of the extension $\Bbb{Q}(\zeta_n)/\Bbb{Q}$ is $\varphi(n)$ where $\varphi$ is the Euler Totient Function. Now it is not hard to see that the only values of $n$ for which $\varphi(n) = 2$ is when $n = 2,3,4$ and $6$.
Now when you look at $\Bbb{Q}(\sqrt{-2})$ and $\Bbb{Q}(\sqrt{-3})$, if you have an $n^{th}$ root of unity in there it can only be for those stipulated values of $n$ above, because otherwise you have a $\Bbb{Q}$ - subspace of dimension greater than 2 sitting inside of a $\Bbb{Q}$ - vector space of dimension 2 which is impossible. Now let us write out $\zeta_n$ for these values of $n$, we have: $\zeta_2 = - 1$, $\zeta_3 = \frac{-1 + \sqrt{3}i}{2}$, $\zeta_4 = i$ and $\zeta_6 = \frac{1 + \sqrt{3}i}{2}.$ 
Can you now complete your problem? I leave the rest for you since this is a homework problem. By applying degree arguments, etc. you should be able to eliminate cases. For example, you should be able to work out for yourself why $\pm i$ is not in $\Bbb{Q}(\sqrt{-3})$ say.
