lean_workbook_plus_33584

$\zeta = \cos \frac {2\pi}{n} + i\sin \frac {2\pi}{n} = e^{\frac {2\pi i}{n}}$ , and thus ${ \{\zeta^k\}}$ are the $n^{th}$ roots of unity, for all $1\leq k \leq n - 1$ , and therefore the roots of $x^n = 1 \implies x^n - 1 = 0$ . The fundamental theorem of algebra allows us to factor it as $x^n - 1 = (x - 1)(x - \zeta)(x - \zeta^2)...(x - \zeta^{n - 1})$ .