Prove the "strange" isomorphism: $\mathbb C^*/C_n\simeq \mathbb C^*$.
My solution: The map $f:x\mapsto x^n$ is homomorphism since $f(x)f(y)=x^n y^n=(xy)^n=f(xy)$, $\mathrm{ker}\,f=C_n$, $\mathrm{im}\,f=\mathbb C^*$. Therefore, by the homomorphism theorem: $\mathbb C^*/C_n\simeq \mathbb C^*$.
Here $C_n$ is the root of unity. Could someone prove it and correct if it's needed.