Proposition 1

The topological group $\mathbf{C}^*$ is isomorphic to the product of the topological groups $\mathbf{R}^*$ and $\mathbf{U}$.

For the mapping $z \to \left( |z|, \frac{z}{|z|} \right)$ is a homeomorphism of $\mathbf{C}^*$ onto $\mathbf{R}_+^* \times \mathbf{U}$ (Chapter VI, § 2, no. 3, Proposition 3); and it follows immediately that it is an isomorphism of the group structures.

The topological group $\mathbf{R}_+^*$ is already known to be isomorphic to the additive group $\mathbf{R}$ (Chapter V, § 4, Theorem 1); the study of the topological group $\mathbf{C}^*$ is therefore reduced to that of $\mathbf{U}$, which we shall consider in § 2.