Originally Posted by
robeuler I am doing a series of exercises with the goal of proving that the automorphism group of R/Q is trivial.
I have proven that any automorphism, g, takes positive reals to positive reals.
I haven't made progress on any of the following steps:
b) If a<b then g(a)<g(b) for any real numbers a and b.
c) Prove that -1/m<a-b<1/m implies that -1/m<g(a)-g(b)<1/m for every positive integer m.
d) It follows from c) that g is continuous (this is clear to me)
e) And finally prove that any continuous map on R which fixes Q is the identity map.
I think I know how to do e). Since Q is dense in R, for any real r pick 'nearby' rationals p and q such that p<r<q. Since p and q are fixed by g, and g(r) has to be between p and q, we know that g is the identity, because we can pick p and q as close to r as we like.