Is the group $S^1:=\{z\in \mathbb C :|z|=1\}$ isomorphic with $S^1\times S^1$ ?

Question

Is the group $S^1:=\{z\in \mathbb C :|z|=1\}$ isomorphic with $S^1\times S^1$ ?

I know that $\mathbb C^* \cong S^1 \cong \mathbb R/\mathbb Z \cong \mathbb C^*/\mathbb R^+$ , but I cannot make any headway towards the original problem . Please help . Thanks in advance

Answer 1

There are $n$ elements which satisfies $x^{n}=1$ in $S^{1}$ ($n$-th root of unities), but $n^{2}$ elements in $S^{1}\times S^{1}$ ($(\zeta_{n}^i, \zeta_{n}^j)$ for $1\leq i, j\leq n$).