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

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!

Hint: $Q^\times$ is isomorphic to $\Bbb Z_2+\sum_{n\in\Bbb N}\Bbb Z$.
• Well I haven't found the auutomorphisms. The reason $Q^\times$ is isomorphic to that sum is that if $r$ is a non-zero rational then $r$ has a unique prime factorization $r=\pm p_1^{n_1}p_2^{n_2}\dots p_N^{n_N}$. – David C. Ullrich Mar 11 '18 at 16:09
• I know this fact, so the question is how to send primes. It not necessarily send primes to integral numbers, moreover, it’s not reasonable to arrange primes at random. Say, send 2 to $1/2$) ,3 to $1/3$ and 5 to $1/6$ , but here troubles come, since this map must sent 6 to $1/6$. So I can’t see a reasonable way. – Jiabin Du Mar 11 '18 at 16:41
• Any element of order two must map to an element of order two, so you can forget about the $\Bbb Z_2$. For a warmup, try finding the automorphisms of $\Bbb Z^2$... – David C. Ullrich Mar 11 '18 at 17:00