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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?

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up vote 8 down vote accepted

Rough idea: Write $x\in\mathbb{R}$ as $x=\sigma t$ where $\sigma=\pm 1$ and $t=|x|$. Then apply the log idea to the $t$ part; the $\sigma$ part will behave like the integers mod 2.

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