# Algebraic structure of the roots of a complex number

Let $$n$$ be any integer number, and let $$w$$ be a non-zero complex number. In polar form, let us suppose $$w = r\exp(i\phi)$$. The solutions of the equation $$z^n = w$$ are therefore $$\alpha, \quad \alpha\omega, \quad \alpha\omega^2, \quad \dots, \quad \alpha\omega^{n-1}$$ where $$\omega$$ is the $$n$$th complex root of unity with the smallest positive argument, and where $$\alpha = \sqrt[n]{r}\exp\left(\frac{i\phi}{n}\right)$$ Using de Moivre's theorem, it is easy to prove this fact. In fact, writing $$z = \rho \exp(i\theta)$$ and substituting this into $$z^n = w$$ gives $$\rho^n\exp(in\theta) = r\exp(i\phi)$$ and therefore $$\rho = \sqrt[n]{r}, \qquad n\theta \equiv \phi \pmod{2\pi}$$ and the second of these equations gives $$\theta = \frac{\phi}{n} + \dfrac{2k\pi}{n} \quad k = 0,1,2,\dots,(n-1)$$ so eventually $$z = \sqrt[n]{r}\exp\left(\frac{i\phi}{n} + \frac{2i\pi}{n}\right) = \sqrt[n]{r}\exp\left(\frac{i\phi}{n}\right)\exp\left(\frac{2k\pi i}{n}\right) = \alpha\omega^k$$ The proof above is okay, but I was wondering whether there was a more 'algebraic' proof of this fact. After all, it is sort of considering $$z^n = w$$ as $$z^n = w \times 1$$ and extracting the $$n$$th root, one would get $$z = \sqrt[n]{w}\times\sqrt[n]{1}$$, but this is not formal. What I was looking for is whether there is a purely algebraic reason of this fact, that doesn't involve the polar form of a complex number. After all, the set $$\mathbb{U} = \{ z \in \mathbb{C} : |z|=1\}$$ is rich of structure.

Thank you for any suggestion you may give me.

Given any two roots, $$z_1,z_2$$ you have $$\left(\frac{z_1}{z_2}\right)^n =\frac ww=1.$$ So, once you have one solution, the other $$n$$ can be expressed as the product of that solution and some solution to $$y^n=1.$$
You still need to find one root of $$z^n=w,$$ but I don’t think there is an algebraic solution there.
Hint: If $$z_1$$ and $$z_2$$ are solutions of $$z^n=w$$, then $$z_1/z_2$$ is a solution of $$z^n=1$$ and so $$z_1=z_2 \omega$$, where $$\omega$$ is a $$n$$th root of unit.