## 9367 Q51

Forum for the GRE subject test in mathematics.
yoyobarn
Posts: 80
Joined: Sun Dec 19, 2010 7:01 am

### 9367 Q51

Hi, what are the two distinct automorphisms?

I think one of them is the identity map: phi(n)=n

Thank you very much for help.

cincodemayo5590
Posts: 11
Joined: Tue Jan 24, 2012 6:43 pm

### Re: 9367 Q51

The only field automorphism $$\phi : \mathbb{Q} \rightarrow \mathbb{Q}$$ is the identity. This can be seen since $$\phi(1) = \phi(1) \phi(1) \implies \phi(1) = 1$$ or $$\phi(1) = 0$$, but the latter does not give an injective map, since $$\phi(0) = \phi(0) + \phi(0) \implies \phi(0) = 0$$; then, $$\phi(n) = n \phi(1)$$ for integers $$n$$; and thus, for integers $$p$$, $$q$$, since $$\phi(q) \phi \left( \frac{p}{q} \right) = \phi(p)$$, we have $$\phi \left( \frac{p}{q} \right) = \frac{ \phi(p)}{\phi(q)} = \frac{p}{q}$$.

Is there a solution manual that says there are two field automorphisms on the rationals? That's wrong.

aaroncraig
Posts: 7
Joined: Thu Feb 16, 2012 2:07 pm

### Re: 9367 Q51

The answer key at the end has "B" (i.e. there is exactly one automorphism) as the answer

yoyobarn
Posts: 80
Joined: Sun Dec 19, 2010 7:01 am

### Re: 9367 Q51

Thanks for the detailed explanation!

Sorry my bad, I mentally associated "B" with 2..

DanielMcLaury
Posts: 12
Joined: Mon Nov 21, 2011 10:42 pm

### Re: 9367 Q51

Expanding on this, nontrivial field automorphisms have to fix some proper subfield, but the rationals don't contain subfields -- $$\mathbb{Q}$$ its own prime subfield (which is essentially what cincodemayo5590 shows above).