The complex number $\omega$, where $\omega \neq 1$, is a root of the equation \(z^3 - 1 = 0

Question

The complex number $\omega$, where $\omega \neq 1$, is a root of the equation $z^3 - 1 = 0$. Show that $1 + \omega + \omega^2 = 0$. Hence show that $1 + \omega^k + \omega^{2k}$, where $k$ is a positive integer, is not zero when $k$ is a multiple of $3$, but zero when $k$ is not a multiple of $3$.

Accepted Answer

(a) $1+\omega+\omega^2=0$. (b) The sum is $3$ when $3\mid k$, and $0$ otherwise.

Geometric sum of roots of unity

Geometric sum of roots of unity

Solution:

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