# Automorphisms of Q

November 2, 2011 Leave a comment

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 (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 be an automorphism of . That is, let . Let be a rational number. Then we can write as

If we apply ,, we get that

This is to say, any field automorphism of a field extending fixes . Another way to say this is that we can reach any number in 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!