# Prove the direct product of nonzero complex numbers under multiplication.

Let $\mathbb{C}^{\times}$ be the group of nonzero complex numbers under multiplication. Then $\mathbb{C}^{\times}$ is the direct product of the circle group $T$ of unit complex numbers and the group $\mathbb{R}^{+}$ of positive real numbers under multiplication.

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Define $f:{\mathbb C}^{\times}\to T\times\mathbb{R}^{+}$ by $f(z)=(\frac{z}{|z|},|z|)$ and prove that this is an isomorphism of groups.

Actually $\mathbb{C}^{\times}$ is the internal direct product of the two since ${\mathbb C}^{\times}=T\mathbb{R}^{+}$ (write $z=\frac{z}{|z|}\cdot|z|$) and $T\cap\mathbb{R}^{+}=\{1\}$.

Hint: take the following “other-direction” map as that of YACP:

$$g: \Bbb T\times \Bbb R^+\to \Bbb C^*\;;\;\;g(e^{it},r):=re^{it}\;,\;\;t\in\Bbb R$$

i.e., the polar-coordinates representation for a non-zero complex number.