summing the nth root of complex numbers (NOT unity)

Question

I have to sum the n nth roots of any complex number, to show = 0. This question does not specify unity, and every other proof I can find is only in the case of unity. My current thoughts are trying to make a geometric sum with powers of 1/n, but I can't justify this =0. I would be so grateful for any advice you could give me, or any alternative solutions to reach my answer. Thank you

Answer 1

Write an arbitrary $n^{th}$ root of $a$ as $\sqrt[n]{a} = |\sqrt[n]{a}|e^{2\pi i k/n}$. Then summing all such $n^{th}$ roots, we get

$$|\sqrt[n]{a}|(1+e^{2\pi i /n}+\ldots + e^{2\pi i (n-1)/n})=|\sqrt[n]{a}|\cdot 0 =0.$$

Answer 2

Hints:

(1) If $\;a_0+a_1x+\ldots+a_nx^n\;$ is a polynomial over some field and its roots (perhaps with repetitions) in some other bigger field are $\;c_1,...,c_n\;$ , then

$$c_1+c_2+\ldots+c_n=-\frac{a_{n-1}}{a_n}$$

(2) The $\;n\;\;n\,-$ th roots of a complex number $\;w\;$ are the roots of $\;z^n-w\;$ .