How to prove that $\mathbb{C}^\times \not\simeq \mathbb{R}^\times \times S^1$.

Question

I want to prove that $\mathbb{C}^\times \not\simeq \mathbb{R}^\times \times S^1$ (even though $\mathbb{C}^\times\!/\mathbb{R}^\times \simeq S^1$). I know that $\mathbb{C}^\times$ is not an internal direct sum of $\mathbb{R}^\times$ and $S^1$. But that does not imply that the external direct sum is not isomorphic to $\mathbb{C}^\times$. For example, $W_1 = \{ (x,y,0) : x,y\in \mathbb{R} \}$ and $W_2 = \{ (x,0,0) : x \in \mathbb{R}\}$ are subspaces of $\mathbb{R}^3$ and $W_1 \simeq \mathbb{R}^2$ and $W_2 \simeq \mathbb{R}$ so $W_1 \times W_2 \simeq \mathbb{R}^3$, but $\mathbb{R}^3 \neq W_1 \oplus W_2$. So, can someone give me a proper argument showing that $\mathbb{C}^\times\not\simeq \mathbb{R}^\times \times S^1$?

Answer 1

$\mathbb C^\times$ has two square roots of unity: $1$ and $-1$.

$\mathbb R^\times \times S^1$ has four square roots of unity: $(1, 1)$, $(1,-1)$, $(-1,1)$, and $(-1,-1)$.