# nth root of unity problem for homework

I'm super confused right now with this problem and any kind of mini lecture would be awesome. Thank you

Let $ω = e^{2πi/n}$ , where $n$ is a positive integer. Prove that

(a) $1 · ω · ω^2 · · · ω^{n−1} = (−1)^{n−1}$ .

• Hint: $0+1+2+\dots+n-1 = n(n-1)/2$ – GNUSupporter 8964民主女神 地下教會 Feb 19 '18 at 21:53
• Thanks for the comment. your hint helped – Tony Mau Feb 19 '18 at 22:09
• Welcome to Math.SE! To avoid downvotes in future questions, please read this post and the others there for information on writing a good question for this site. In particular, people will be more willing to help if you include some motivation, and an explanation of your own attempts. – GNUSupporter 8964民主女神 地下教會 Feb 19 '18 at 22:33

$$1\cdot\omega\cdot\omega^2\cdots\omega^{n-1}=\omega^{1+2+\ldots+(n-1)}$$
Using that $e^{\pi i}=-1$, and that $1+2+\ldots+(n-1)=\frac{n(n-1)}{2}$, the product becomes $$\omega^{\frac{n(n-1)}{2}}=e^{\frac{2\pi i}{n}\cdot\frac{n(n-1)}{2}}=(-1)^{n-1}$$