I have recently stumbled upon a problem which says:
Given that $z^5-1=0$, find all solutions for $z\in\mathbb{C}$.
I have researched this problem a bit, and I have discovered that this has something to do with a Root of Unity. What is a root of unity and how do I solve one? Also, is there a formula for finding all complex solutions of $z^n=1$, where n is a constant? If so, what does this have to do with a root of unity?
EDIT
Another question: I see the $\cos(x)+i\sin(x)$ formula for all $z^n=1$, but is there a formula for all $z^n=c$ for some $c \in \mathbb{R}$, or at least positive $c \in \mathbb{R}$?
Best Answer
Definition (Root of unity). An element $\zeta$ of a ring $R$ with unity $1$ such that $\zeta^m = 1$ for some $m \ge 1$. The least such $m$ is the order of $\zeta$. -Encyclopedia of Mathematics
In the field of complex numbers, roots of unity of order $m$ take the form $$\cos\frac{2\pi k}{m} + i \sin\frac{2\pi k}{m},$$ where $k=0,1,\dots,m-1$.
We may write $1$ in polar form as $e^{2\pi i k}$ and therefore $\zeta^m=e^{2\pi i k}\Rightarrow\zeta=e^{\frac{2\pi i k}{m}}$. With Euler's formula, we can decompose the exponential into real and imaginary parts to get our final result, which is stated above. If we add the condition that $k$ must be relatively prime to $m$, we only get the so-called primitive roots.
The same can be done with $\zeta^m=c,$ where $c$ is a real number. Write $c$ as $ce^{2\pi i k}$ to get the slightly generalized result $\zeta=\sqrt[m]{c}\,(\cos\frac{2\pi k}{m} + i \sin\frac{2\pi k}{m})$.
Complex roots of unity 5 $\qquad\qquad\qquad\qquad\:\:$ The mapping $z\mapsto z^5-1$