Product-of-n-n-th-roots-of-unity-1-2-3-n-1-1-n-1-Why-How-to-get-RHS-

Question Number 19312 by Tinkutara last updated on 09/Aug/17

Product of n, n^{th} roots of unity

= 1 . α . α^{2} . α^{3} \dots . . α^{n - 1} = {(- 1)}^{n - 1}

Why ? How to get RHS ?

Answered by ajfour last updated on 09/Aug/17

x^{n} - 1 = (x - 1) (x - α) (x - α^{2}) \dots (x - α^{n - 1})

comparing constant term on

each side :

- 1 = {(- 1)}^{n} α^{1 + 2 + \dots . + (n - 1)}

\Rightarrow 1 . α . α^{2} . α^{3} \dots α^{n - 1} = {(- 1)}^{n - 1} .

Commented by Tinkutara last updated on 09/Aug/17

Thank you very much Sir!