For which rational angles (I call an angle $\alpha$ rational if $\alpha/\pi \in \mathbb Q$) is $\sin(\alpha)$ algebraic with $[\mathbb Q(\alpha):\mathbb Q]$ being a power of 2? This condition is equivalent to $\alpha$ being the result of a straightedge and compass construction.
While writing down my approach I think I already solved the question. I think the answer is as follows:
Let $N$ be a square free product of Fermat primes then $\alpha=\pi \cdot n/(2^m\cdot N)$ has the desired property for all $n,m \in \mathbb Z$. On the other hand every angle with the desired property can be written in that form.
Is that correct? I guess the reason is because the problem can be transfered into constructing a regular $n$-gon and this problem is solved by the Fermat primes.