How to calculate the automorphisms group of nonzero rational number multiplicative group?

Question

For the nonzero rational number multiplicative group $Q^\times$, how to calculate the automorphisms group $Aut(Q^\times)$ ?

1. First,suppose $\phi :Q^\times \to Q^\times$ is an automorphism,it must send 1 to 1, and -1 to -1, I think the question is to determine the primes to be sent what?
2. But I have trouble in determining this thing. I guess this group is $ Z_2\oplus \oplus_{p \ primes} Z $. Any help will be greatly appreciated, thanks!

Answer 1

Hint: $Q^\times$ is isomorphic to $\Bbb Z_2+\sum_{n\in\Bbb N}\Bbb Z$.