Let $m$ be a positive integer.
I need the existence of a primitive $m$-th root of unity $\zeta_m$ such that its imaginary part is strictly greater than $1/2$.
We can write $\zeta_m = \exp(2\pi i a/m)$ for some $a$ coprime to $m$.
The condition above boils down to $\sin(2 \pi a /m ) > 1/2$. This just means that $ \frac{m}{12} < a < \frac{5m}{12}.$ So I'm looking for the existence of an integer $a$ coprime to $m$ such that $ \frac{m}{12} < a < \frac{5m}{12}. $
Is this always possible?
Probably I need that $m> 12$. For $m=12$, there is no such $a$. It's ok if it doesn't work for a finite number of $m$.