I am trying to learn about Kummer Rings, and in particular what makes $n=3,4,6$ so special. (That is the Gaussian and Eisenstein integers)
The only $\theta\in [0,\frac{\pi}{2}]$ which are rational multiples of $\pi$ for which $\cos(\theta)\in \mathbb{Q}$ are $\theta=\frac{\pi}{2},\frac{\pi}{3}$ which corresponds exactly to $n=4,6$ in $\frac{2\pi}{n}$.
Can someone give me an explanation for why $\cos(\theta)$ is rational only in these cases? Also, can we go the other way, and use some nice property of the Kummer Rings to show that $\cos(2\pi/n)$ is rational if and only if $n=1,2,3,4,6$?
Thanks,
Edit: As pointed out by Qiaochu, what I previously wrote above was certainly not the norm.