1.19

1.19
Fix a positive integer \(n\) and a complex number \(w\). Find all solutions to \(z^n=w\).
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Let \(n \in \mathbb{Z}\) and \(z,w \in \mathbb{C}\). In polar coordinates, \(w=re^{i\theta}\) so: \(z^n=re^{i\theta}\). Let \(z=me^{i\phi}\). The modulus of \(z\) is \(m^n=r\) ie \(m=\sqrt[n]{r}\). So for \(\theta\) of \(z\), we have \(n\theta=\theta+2\pi k\) for some \(k \in \mathbb{Z}\) and \(\theta=\frac{\theta+2 \pi k}{n}\). So \(z=\sqrt[n]{r}*e*\frac{i(\theta+2 \pi k)}{n}\). By the fundamental theorem of algebra we know that there are \(n\) solutions to this equation. Restricting \(k\) so \(0 \leq k \leq n\), then there are \(n\) unique solutions for \(z\).