prove that $\Bbb Z/n \Bbb Z \cong \mu_n$

Question

I need to prove that $\Bbb Z/n \Bbb Z \cong \mu_n$

$\Bbb C^x \gt \mu_n = \{z \in \Bbb C^x | z^n = 1 \}$

what i tried - I tried building a homomorphism $f: \Bbb Z \to \mu_n$

such that $f(z) = e^{{2 \pi iz}/n}$

and then say that Ker(f) = n$\Bbb Z$

but I get that $f(z_1z_2) \ne f(z_1)f(z_2)$

any help will be appriciated

Answer 1

As others have said, your homomorphism is correct you just mixed up the operation. Note

$$f(z_1+z_2)=e^{2\pi i(z_1+z_2)/n}=e^{2\pi iz_1/n}e^{2\pi iz_2/n}=f(z_1)f(z_2)$$

Now you just need to show it's surjective and you're done.