Got this question on a recent exam, and though it may seem trivial, I cannot seem to figure it out.
Show that $\mathbb{R}^{*} \cong \mathbb{R} \times \mathbb{Z}/2\mathbb{Z}$.
I had one of these questions in the past, however the underlying set of multiplicative $\mathbb{R}$ was given to be only positive (\mathbb{R}_{>0}) making it easy to define the isomorphism using a logarithm.
I can see intuitively that the integers modulo 2 group is ment to preserve whether the input is even or uneven, however I cannot figure out a suiting isomorphism. Any tips?