What is the Sum of nth Roots of Unity and How Can It Be Proven?

[converting1]

i'm trying to prove the sum of nth roots of unity = 0, but I don't really know how to proceed:

suppose z^n = 1 where z ε ℂ,

suppose the roots of unity for z are 1, ω, ω^2, ω^3 ... ω^n

the sum of these would be S = 1 + ω, ω^w, ω^3 +...+ ω^(n-1) + ω^n

from here I had an idea to do some fancy manipulation of S, then show that S = 0, but if say I do ωS - S I don't get 0!

I'm assuming I've made a very silly mistake or the way of approaches this is all wrong,

does anyone have a better approach or can anyone spot my mistake?

thanks,

 

[Mark44]

i'm trying to prove the sum of nth roots of unity = 0, but I don't really know how to proceed:

suppose z^n = 1 where z ε ℂ,

suppose the roots of unity for z are 1, ω, ω^2, ω^3 ... ω^n

That's too many. Your roots should start at exponent 0 and end at exponent (n - 1).

the sum of these would be S = 1 + ω, ω^w, ω^3 +...+ ω^(n-1) + ω^n

S = 1 + ω + ω² + ... + ω^{n - 1}

Notice that the right side is a (finite) geometric series.

from here I had an idea to do some fancy manipulation of S, then show that S = 0, but if say I do ωS - S I don't get 0!

I'm assuming I've made a very silly mistake or the way of approaches this is all wrong,

does anyone have a better approach or can anyone spot my mistake?

thanks,

 

[Dick]

And if you make the correction Mark44 gave, you will find ωS - S is zero.

 

[tiny-tim]

alternatively, they're the n roots of the polynomial equation xⁿ - 1 = 0 …

so which coefficient is the sum of the roots? ​