Automorphisms of Q

We’ve talked a bit about automorphisms, but we haven’t seen very much of them, so I wanted to do an example. What is \mbox{Aut}(\mathbb Q) (the group of automorphisms of the field of rational numbers)?

It definitely contains the identity (a.k.a. lame) automorphism, since every field has that one. Is there anything else? No, and here is why. Let \phi be an automorphism of \mathbb Q. That is, let \phi\in\mbox{Aut}(\mathbb Q). Let p/q be a rational number. Then we can write p/q as

p/q=\frac{\overbrace{1+\cdots+1}^{p\ times}}{\underbrace{1+\cdots+1}_{q\ times}}

If we apply \phi,, we get that

\phi(p/q)=\frac{\overbrace{\phi(1)+\cdots+\phi(1)}^{p\ times}}{\underbrace{\phi(1)+\cdots+\phi(1)}_{q\ times}}=\frac{p\cdot\phi(1)}{q\cdot\phi(1)}=p/q.

This is to say, any field automorphism of a field extending \mathbb Q fixes \mathbb Q. Another way to say this is that we can reach any number in \mathbb Q just from 1 and operations which any field automorphism respects (namely addition and division).

This is to say, the automorphism group of the field of rationals is trivial!


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