I am looking for a general solution for the roots of unity for a fractional exponent. That is to say, complex values of $z$, such that $z^n=1$ for $n=\frac{p}{q},p,q\in \Bbb{Z}, p$ is coprime to $q$.
So far, I have tried the usual approach of taking $\frac{2\pi k}{n}$, for $k \in \{1,2,...,n \} $. This works for the integer values of $n$, but when applied to a non-integer $n$, it is unclear what the range of $k$-values should be.
Like for integer values of $n$, the possible roots of unity for non-integer $n$-values is periodic, but produces far too many extraneous roots, which, when raised to the $n$th power, don't give $1$.
Using WolframAlpha, it appears that the (non-trivial) solutions it gives for $z^n=1$ (for non-integer $n$) are the "left-most" solutions on the complex plane, i.e. the most negative. These come in conjugate pairs, and seem to yield $m$ solutions (including $1$), where $m$ is the nearest odd number to $n$.
I am struggling to find a link between the position on the complex plane of these roots, and a general formula for the roots of unity. I eventually hope to extend this to irrational $n$, or even non-real $n$, so any hope would be greatly appreciated.