Date: Fri, 21 Nov 1997 17:51:22 -0600 (CST)
From: Dave Rusin <rusin>
To: gwsmith@gwi.net
Subject: Re: David Ullrich's challenge: define the field C
Newsgroups: sci.math

In article <654he0$kog$1@noc1.gwi.net> you write:
>Pertti Lounesto (lounesto@torstai.hit.fi) wrote:
>
>: 2. Neil Rickert learned that R has only one automorphism, the identity.
>
>R has only one continuous automorphism. It has a lot of automorphisms
>as an extension of Q; Gal(R/Q) is huge.

If  f: R -> R  is a nonzero homomorphism of rings, then  f(x)=x for all  x  in Q.

Also, y>0 => y=x^2 for some real x => f(y)=f(x)^2 => f(y)>0.
Then y>z => y-z>0 =>f(y)-f(z)=f(y-z)>0 => f(y) > f(z).

So  f  a nonzero homomorphism => f preserves  Q  and preserves order, so
it's the identity on all of  R.

[deletia -- djr]

dave