Multiplicative Group of Complex Numbers is not Isomorphic to Multiplicative Group of Real Numbers

Theorem
Let $\struct {\C_{\ne 0}, \times}$ be the multiplicative group of complex numbers.

Let $\struct {\R_{\ne 0}, \times}$ be the multiplicative group of real numbers.

Then $\struct {\C_{\ne 0}, \times}$ is not isomorphic to $\struct {\R_{\ne 0}, \times}$.

Proof
$\struct {\C_{\ne 0}, \times}$ is isomorphic to $\struct {\R_{\ne 0}, \times}$.

Let $\phi: \C_{\ne 0} \to \R_{\ne 0}$ be an isomorphism.