Find all solutions to:
$\displaystyle z^i=1$
Can someone give me the method to solve this?
The equation rewrites $\displaystyle e^{i \log z}=1$ (which may depend on the determination of the complex logarithm), so you are first reduced to solving $\displaystyle e^{w}=1$ for $\displaystyle w\in\mathbb{C}$, which is a classic one (decompose w in real and complex parts if you don't know the answer already).
You'll find that the final answer actually doesn't depend on the determination of the logarithm.