Hello,

I want to factorize the equation Z^n=1 over complex numbers

I reached (x-1)(X^4+X^3+X^2+x+1) andd then got stucked

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- Sep 16th 2011, 09:50 AMHyunqulComplex numbers Factorization
Hello,

I want to factorize the equation Z^n=1 over complex numbers

I reached (x-1)(X^4+X^3+X^2+x+1) andd then got stucked - Sep 16th 2011, 10:10 AMTheChazRe: Complex numbers Factorization
Hmm

X

x

Z

n (=5)?? - Sep 16th 2011, 10:22 AMProve ItRe: Complex numbers Factorization
First of all, I don't know where x and X have come from, since your equation is in terms of z...

Anyway, let , then

Therefore

The solutions are all evenly spaced about a circle, so all have the same modulus and are separated by an angle of , so the solutions are

Each solution implies a factor of , so that means the factorised form (finally) is

- Sep 16th 2011, 12:40 PMPlatoRe: Complex numbers Factorization
Here is a compact way to factor .

Let then write - Sep 16th 2011, 09:10 PMProve ItRe: Complex numbers Factorization
- Sep 17th 2011, 05:11 AMHallsofIvyRe: Complex numbers Factorization
The nth roots of unity, z such that are given by such that so we must have r= 1, which is, of course, equivalent to . That is, geometrically, the n nth roots of unity lie on the unit circle in the complex plane, equally space around the circle. The five fifth roots of unity, in particular, form a "pentagon" on the unit circle.