I want to solve $z^n=1$ in the complex field.
We can express $z$ in standard form or in polar form. Which one is better? I guess that it is polar form as there are nice theorems about multiplication of complex numbers in polar form, so I will adopt this form.
Using DeMoivre theorem:
$z^n = r^n[cos(nA)+isin(nA)]$
So, we want to solve $z^n = r^n[cos(nA)+isin(nA)]$ = 1
I am stuck here.
All I know, by some reasoning is that $isin(nA)$ must be $0$, and so, $nA$ must be either $0º$ or $180º$ (up to congruences), and, therefore, $cos(nA)$ must be either $1$ or $-1$. But what, about $r$? It could be either $1$ or $-1$?
I am having a big headache right now, so maybe there are typos here I did not perceive even after reading it all two times... Could somebody clarify this mess to me?