Given the complex polynomial equation $z^{n} -1=0$  for n=2, 3, 4, what are the corresponding roots?

Given the complex polynomial equation z^{n} -1=0z n ?1=0 for n=2, 3, 4, what are the corresponding roots?

Answer

For $n=2$, the roots are $1$ and $-1$. For $n=3$, the roots are $1$, $\frac{-1+i\sqrt{3}}{2}$, and $\frac{-1-i\sqrt{3}}{2}$, where $i^2 = -1$. In fact, \[ \left(\frac{-1 \pm i\sqrt{3}}{2}\right)^3 = \frac{1}{8}(\pm (i\sqrt{3})^3 - 3(i\sqrt{3})^2 \pm 3(i\sqrt{3}) - 1) = \frac{1}{8} (\mp 3i \sqrt{3} + 9 \pm 3i \sqrt{-3} - 1) = 1. \] For $n=4$, the roots are $1, i, -1, -i$, where $i^2 = 1$, and the check is immediate, as $\sqrt{-1}^4 = (-1)^2 = 1$.

The answer is accepted.