roots of $z^a=1$, where $a\in \mathbb{R}$ and $z\in \mathbb{C}$

Question

Let $a$ be a positive real number and $z$ be a complex number. I want to find the roots of the equation $z^a=1$. When $a$ is positive integer, say $a=n$, I know that it has $n$ roots but what is the number of the roots of the equation $z^a=1$ when $a\in \mathbb{R}$.

Answer 1

Let $z=r\,e^{i\theta}$, $0\le\theta<2\,\pi$. You wantto solve $$ z^a=r^a\,e^{ia\theta}=1=e^{2k\pi i},\quad k\in\mathbb{Z}. $$ Then $$ \theta=\frac{2\,k\,\pi}{a},\quad k\in\mathbb{Z}. $$ Two solutions corresponding to $k_1,k_2$ are equal if $$ \frac{2\,k_1\,\pi}{a}-\frac{2\,k_2\,\pi}{a}=2\,m\pi $$ for some integer $m$, that is, if $k_1-k_2=a\,m$. If $a$ is irrational then all solutions are different.