prove that every automorphism of R*, the group of nonzero real number under mult

Question

prove that every automorphism of R*, the group of nonzero real number under multiplication , maps positive numbers to positive numbers and negative numbers to negative numbers.

Explanation / Answer

Call your automorphism f. Suppose x > 0. Then there is some nonzero real number r such that r^2 = x. Now apply the automorphism to get ( f(r) )^2 = f(r^2) = f(x). Therefore f(x) > 0. Conversely, suppose f(x) > 0. Applying the same argument with the automorphism f^-1 instead of f, we get x > 0. The result then follows from the fact that every element of R* is either positive or negative, but not both at the same time.

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