Why are roots of unity so important? Can someone give a somewhat elementary explanation as to why roots of unity is so important? (I only know up to elementary number theory/group theory.)
Also:


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*Do similar roots of unity type structures exist for other closed curves such as an ellipse? I was thinking what'd happen to algebraic description of the roots of unity if you deform the circle into some other shape.

*It seems like linear relations of roots of unity are well studied. What's the motivation for studying that? What about non-linear relations?

*Are there any open problems regarding roots of unity? 
 A: Algebraic Number theory studies numbers  that are roots of polynomial equations (in one variable) having integer coefficients, and also fields generated by them if you do not know   what a field is, important  examples are  simply  subsets of complex numbers containing $0,1$ and closed under all four basic arithmetic operations: $+,-,\times, \div$).
Among all polynomial equations $x^n-1$ are simple in nature. The fields generated by them (cyclotomic fields) have abelian Galois groups. Galois group members are automorphisms which are functions, and when a group of functions satisfy commutative law for composition it is significant. Kronecker-Weber theorem states that the cyclotomic fields and their subfields are the only examples over  rational number fields to have abelian Galos group.
And Kronecker's jugendtraum is about generalisation for other fields, it is not  going from circle to  ellipse, but in a different way.
About non-linear relations on roots of unity: one simply takes  the minimal polynomial satisfied a root of unity. If several roots of unity are there take the one whose order is the lcm of the orders of all the given  roots of unity then all of them are powers of that number.
A: There are a few things to say about this. Consider $\zeta_p = e^{\frac{2 \pi i}{n}}$, a primitive $n^{th}$ root of unity. Then the $n^{th}$ cyclotomic field, $\Bbb Q(\zeta_n)$ has interesting properties. Here is one of my favorite examples. Fix $p$ an odd prime,  $p \equiv 1 \pmod{4}$ and consider $\Bbb Q(\sqrt{p})$, which has only two symmetries, the one sending $\sqrt{p}$ to $\sqrt{p}$ and the other sending $\sqrt{p}$ to $-\sqrt{p}$. So its symmetry group has only $2$ elements and is visibly abelian. There is a theorem from a subject called class field theory that since this field has an abelian symmetry group, that this field should actually be contained inside one of these cyclotomic fields $\Bbb Q(\zeta_m)$ for some $m$. Using some results from algebraic number theory, you can show that in fact $m=p$. Thus, we have the following surprising fact that $\sqrt{p}$ should be a sum of $p^{th}$ roots of unity.
When $p=5$, we have that $\sqrt{5} = \zeta_5 - \zeta_5^2 - \zeta_5^3 + \zeta_5^4$, which is really cool! As far as I know, the expression for general $p$ was noticed by Gauss. The expression (for all odd p) is:
$(G_p)^2 = \left ( \sum_{a=0}^{p-1} \left ( \frac{a}{p} \right) \zeta_p^a\right )^2 = (-1)^{\frac{p-1}{2}}p$
Where $\left ( \frac{a}{p} \right)$ is the Legendre symbol. So when $p \equiv 1 \pmod{4}$, $G_p = \pm \sqrt{p}$, and when $p \equiv 3 \pmod{4}$, $G_p = \pm \sqrt{-p}$. 
It's actually true that the plus sign holds in both of the above, but that is actually very difficult, whereas knowing the equality up to $\pm$ is much easier. 
The moral of the story is, (number) fields with abelian symmetry groups are contained in cyclotomic fields, so to understand them we should study cyclotomic fields. So to answer your question, roots of unity and their sums and products contain a surprising amount of number theoretic information, and is one reason why they're studied. 
