Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

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  ..... α^(n−1)  = (−1)^(n−1)   Why? How to get RHS?

$$\mathrm{Product}\:\mathrm{of}\:{n},\:{n}^{\mathrm{th}} \:\mathrm{roots}\:\mathrm{of}\:\mathrm{unity} \\ $$$$=\:\mathrm{1}.\alpha.\alpha^{\mathrm{2}} .\alpha^{\mathrm{3}} \:.....\:\alpha^{{n}−\mathrm{1}} \:=\:\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \\ $$$$\mathrm{Why}?\:\mathrm{How}\:\mathrm{to}\:\mathrm{get}\:\mathrm{RHS}? \\ $$

Answered by ajfour last updated on 09/Aug/17

x^n −1=(x−1)(x−α)(x−α^2 )...(x−α^(n−1) )  comparing constant term on  each side:  −1=(−1)^n α^(1+2+....+(n−1))   ⇒ 1.α.α^2 .α^3 ...α^(n−1) =(−1)^(n−1)  .

$$\mathrm{x}^{\mathrm{n}} −\mathrm{1}=\left(\mathrm{x}−\mathrm{1}\right)\left(\mathrm{x}−\alpha\right)\left(\mathrm{x}−\alpha^{\mathrm{2}} \right)...\left(\mathrm{x}−\alpha^{\mathrm{n}−\mathrm{1}} \right) \\ $$$$\mathrm{comparing}\:\mathrm{constant}\:\mathrm{term}\:\mathrm{on} \\ $$$$\mathrm{each}\:\mathrm{side}: \\ $$$$−\mathrm{1}=\left(−\mathrm{1}\right)^{\mathrm{n}} \alpha^{\mathrm{1}+\mathrm{2}+....+\left(\mathrm{n}−\mathrm{1}\right)} \\ $$$$\Rightarrow\:\mathrm{1}.\alpha.\alpha^{\mathrm{2}} .\alpha^{\mathrm{3}} ...\alpha^{\mathrm{n}−\mathrm{1}} =\left(−\mathrm{1}\right)^{\mathrm{n}−\mathrm{1}} \:. \\ $$$$ \\ $$

Commented by Tinkutara last updated on 09/Aug/17

Thank you very much Sir!

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com