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
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
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.