Thread: Roots of unity isomorphic to cyclic group?

1. Roots of unity isomorphic to cyclic group?

Hello there!

I've been tasked with showing that A_n, where A_n = {z, exists in the complex numbers | z^n = 1 } is isomorphic to the cyclic group Cn, where Cn is of order n.

My understanding (or memory, rather) of complex numbers is quite dim, so this problem is becoming quite of a hurdle.

2. Originally Posted by charleschafsky
Hello there!

I've been tasked with showing that A_n, where A_n = {z, exists in the complex numbers | z^n = 1 } is isomorphic to the cyclic group Cn, where Cn is of order n.

My understanding (or memory, rather) of complex numbers is quite dim, so this problem is becoming quite of a hurdle.
Hint: The solutions to $z^n = 1$ are $\zeta, \zeta^2, ... , \zeta^n$ where $\zeta = e^{2\pi i/n}$.

3. First note that all the nth roots of unity are powers of $e^{\frac{2 \pi i}{n}}$
Now define a function $f:C_n \rightarrow \mathbb{C}$
where
$f(a)=e^{a\frac{2\pi i}{n}}$

Can you show f is an isomorphism?

4. define an isomorphism from all roots of unity to An. I did this problem before, I forget how the isomorphism went. But it is not hard to show.