this is my first stack exchange post, so forgive my lack of knowledge on formatting
I'm trying to prove that the $n$th roots of unity together with multiplication form a subgroup from $S^1$, where $S^1 = \{e^{i\theta}\;|\; \theta\in\mathbb{R}\}.$
I am stuck in trying to prove that the $n$th roots of unity are closed under multiplication.
If $n$th roots of unity $C_n = \{e^{k(2\pi i/n)}\; |\; k \in \{0,1, \dots, n-1\}\}$
Let $\;a=e^{j( 2\pi i/n)},\; b=e^{\ell(2\pi i/n)}\; |\; 0\leq j, \ell\leq n-1.$
Then $a b = e^{(j+\ell)2\pi i/n}.$ But there's no guarantee that $j+\ell \leq n-1$, so I cannot see how I can claim that $C_n$ is closed under multiplication.
Thank you for any help!
This is what you have to show: if $x$ and $y$ are two roots of unity, then $xy$ is also a root of unity. Since, $x$ is a root of unity, there exists a $n \in \mathbb{N}$ such that $x^n=1$. Similarly, there exists a $m \in\mathbb{N}$ such that $y^m = 1$. Now, you have to find a $k$ such that $(xy)^k=1$. Do you know what that $k$ is?