How can I find fifth root of unity?

Question

I have no idea to do this question, how can I find the fifth root of unity?

Question :

Find all the distinct fifth root of unity. Let $\alpha$ be a fifth root of unity such that $\alpha \ne 1$.

Prove that $1 + \alpha^2 + \alpha^3 + \alpha^4 = 0$.

Your support is much appreciated! thank you.

Answer 1

Here's one: $$ \alpha = \cos(2\pi/5) + \mathbf i \sin (2\pi/5). $$

Now that you know one, can you find a second one, and indeed, the rest of them?

Answer 2

Hint: $~a^5=1\iff a^5-1=0.~$ Now, what formula do you know for $a^n-1$ ? :-$)$

Answer 3

If you can construct, which isn't that difficult with Pythagoras:

$$\cos(36°) = \frac{\sqrt{5} +1}{4}$$

You are done, since $2*36° = 72°$.