Is an automorphism of the field of real numbers R the identity map?
If yes, how can we prove it?
Remark An automorphism of R may not be continuous.
Here’s a detailed proof based on the hint given by lhf.
Let ϕ be an automorphism of the field of real numbers.
Let x>0 be a positive real number.
Then there exists y such that x=y2.
If a<b, then b−a>0.
Hence ϕ(b)−ϕ(a)=ϕ(b−a)>0 by the above.
This means that ϕ is strictly increasing.
If n is a natural number, it can be written in the form 1+…+1, so ϕ(n)=n. Now, any rational number is of the form r=(a−b)c−1, for a,b,c natural numbers, so it follows that ϕ(r)=r for any rational number.
Let x be a real number.
Let r,s be rational numbers such that r<x<s.
Since s−r can be arbitrarily small, ϕ(x)=x.
This completes the proof.