I'm having a hard time with the following question:
Let $S^1=\{z \in \mathbb{C}\mid |z|=1\}$ and $\mathbb{R}_{>0}=\{x\in\mathbb{R}\mid x>0\}$.
Show that $\mathbb{C}^{\times}\cong S^1 \times \mathbb{R}_{>0}$.
I was trying to use Isomorphism theorems, but it didn't work out.
Can I have a subtle hint here? Do I need to construct an isomorphism, or is there a theorem that solves it all?
Thank you.