The $n$th roots of unity are the complex numbers: $1,w,w^2,...,w^{n-1}$, where $w = e^{\frac{2\pi i} {n}}$. If $n$ is even:
The $n$th roots are plus-minus paired, $w^{\frac{n}{2}+j} = -w^j$.
Squaring them produces the $\frac{n}{2}$nd roots of of unity.
Could someone explain the first statement? I understand why the $n$ roots are plus-minus paired (if $n$ is even), but what does the equation mean? An explanation of the equation will be appreciated.
The statement states $w^0$ (or $1$) is a root of unity. Which root is it? Aside from the obvious fact that $1$ is a always a root of $1$, doesn't $w...w^{n-1}$ cover every root? What specific root is $w^0$ referring to?