2.10.5

$$f: \mathbb R^\times \to \{\pm 1\}^\times$$ such that $$f(x) = \begin{cases}1,& x > 0, \\ -1,&\text{ otherwise.}\end{cases}$$

This is a homomorphism with kernel $$P$$, and by the first isomorphism theorem, $$\mathbb R^\times/P$$ is isomorphic to $$\{\pm 1\}^\times.$$