# Solutions to $z^5-1=0$ using "root of unity"?

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 $$c \in \mathbb{R}$$?

• Unity = the number 1 Feb 23, 2021 at 11:50

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 $$z\mapsto z^5-1$$

Given one solution of $$z^n=c$$, the other $$n-1$$ can be found by multiplying by the $$n-1$$ respective $$n$$-th roots of unity other than $$1$$. In general those are the first $$n-1$$ powers of a primitive $$n$$-th root.

So take an example: $$z^5=3$$.

Then take one solution, say $$z=\sqrt[5]3$$. Then if $$\zeta_1,\dots,\zeta_4$$ are the primitive fifth roots of unity, in this case the ones other than $$1$$, we have that the five roots are $$\sqrt[5]3\zeta_1,\dots,\sqrt[5]3\zeta_4, \sqrt[5]3$$.

And note that we have $$\zeta_k=\zeta_1^k$$.

(All of this follows rather easily from the fact that for any $$n$$-th root of unity, the $$n$$-th power is $$1$$. And using Euler's formula: $$e^{i\theta}=\cos\theta+i\sin\theta$$, we also get that $$e^{2\pi i k/n}$$ is primitive iff $$(n,k)=1$$.)