TITLE: Automorphisms of the hyperreals over the rationals and nontrivial automorphism groups
QUESTION [19 upvotes]: A classic result says the automorphism group of $\mathbb{R}$ (over $\mathbb{Q}$) is trivial.  The proof is simple: every automorphism preserves squares, and hence fixes the positive reals, so it must be order preserving.  Since it must fix $\mathbb{Q}$, and $\mathbb{Q}$ is dense in $\mathbb{R}$, if any real number were not fixed, this would yield a contradiction.
In larger real-closed fields where $\mathbb{Q}$ is not dense, automorphisms are still order preserving, but the argument that they are trivial does not work.  I haven't found any examples of a non-trivial automorphism of a real-closed field, but I also can't prove they don't exist.  So, the first question I'd like to ask is whether all automorphism groups of real-closed fields are trivial.
If they aren't all trivial, then I want to know what, if anything, we can say.  To ask a less vague question, can we classify the automorphisms of the hyperreals?  How many are there, and what is their groups structure like?

REPLY [10 votes]: This note complement Joel's, who pointed out that every first order theory with an infinite model has a model with many automorphisms (this was first proved by Ehrenfeucht and Mostowski in their famous 1956 paper on indiscernibles). Joel also outlined why saturated models and countable computably saturated ones have nontrivial automorphisms.  
It is well-known that if $\cal{M}$ is computably saturated and countable, then as soon as we know that some element $m$ of $\cal{M}$ is not first order definable in $\cal{M}$, then we can build an automorphism of $\cal{M}$ that moves $m$. Indeed, we can build continuum-many such automorphisms. This makes for a fun exercise.
Furthermore, it was shown by F. Körner (1998) that if $\cal{M}$ is a countable structure in a countable signature, then there is an elementary extension of $\cal{M}$ that carries an automorphism that is "maximal", i.e., it moves every nonalgebraic element (she showed that $\cal{M}$ can be chosen as any arithmetically saturated elementary extension of M).
In the above, an element $m$ of $\cal{M}$ is said to be algebraic if there is a formula $\phi$ with one free variable such that $\phi(m)$ holds in $\cal{M}$ and $\phi$ has only finitely many solutions in $\cal{M}$.
Körner's theorem was generalized to all structures (whose universe and signature can have any cardinality) by G. Duby in 2003, with a different, more powerful proof technique.
Here are the References:
Friederike Körner, Automorphisms moving all non-Algebraic points and an application to NF, The Journal of Symbolic Logic,Vol. 63, No. 3 (Sep., 1998), pp. 815-830.
Grégory Duby, Automorphisms with only infinite orbits on non-algebraic elements, Archive for Mathematical Logic, July 2003, Volume 42, Issue 5, pp 435–447.