纯数 D — 复数（STEP 3）

For a positive integer $n$, the $n$-th roots of unity are the complex numbers $\omega_k = e^{2\pi i k/n}$ for $k = 0, 1, \ldots, n-1$.

Show that $\displaystyle\sum_{k=0}^{n-1} \omega_k^j = \begin{cases} n & \text{if } n \mid j \\ 0 & \text{otherwise} \end{cases}$ for any integer $j$.

Let $n = 5$ and $\omega = e^{2\pi i/5}$. Show that $1 + \omega + \omega^2 + \omega^3 + \omega^4 = 0$ and hence find $\cos\dfrac{2\pi}{5} + \cos\dfrac{4\pi}{5}$.

Show that $\omega$ satisfies $z^2 + z - 1 = 0$ in some quadratic equation, and hence find the exact value of $\cos\dfrac{2\pi}{5}$.

Using $\cos\dfrac{2\pi}{5}$, find the exact value of $\sin\dfrac{\pi}{5}$ and verify that a regular pentagon inscribed in a unit circle has side length $\sqrt{\dfrac{5-\sqrt{5}}{2}}$.