Proper embedding of a real closed field into another real closed field with the same Archimedean classes Is it true that every real closed field can be elementarily embedded in some other real closed field with the same Archimedean classes (I mean in a proper extension)?
Can for example real numbers be elementarily embedded in another real closed field with the same Archimedean classes? 
R (real numbers) is not the only Archimedean field, is it right?
 A: The question is somewhat ambiguous, it’s not clear whether the Archimedean classes are meant to be additive or multiplicative. I will assume the former, i.e., equivalence classes of the relation
$$a\sim b\Leftrightarrow\mathrm{sign}(a)=\mathrm{sign}(b)\land\exists n\in\omega\smallsetminus\{0\}\,(n^{-1}|a|\le|b|\le n|a|).$$
First, since real-closed fields (rcf) have elimination of quantifiers, any embedding between them is automatically elementary. Thus the question is whether every rcf $R$ has a proper rcf extension $S$ with the same Archimedean classes (i.e., every $s\in S$ is $\sim$ to some $r\in R$).
As Andreas noted above, this property does not hold in general, and in particular, $\mathbb R$ has no proper Archimedean extension. On the other hand, it holds for many other real-closed fields: for example, any Archimedean rcf different from $\mathbb R$ has a proper Archimedean rcf extension (namely, $\mathbb R$). I think the following characterization holds:
Proposition: If $R$ is a rcf, the following are equivalent:


*

*$R$ has a proper rcf extension with the same Archimedean classes.

*There is a Dedekind cut $\langle A,B\rangle$ on the interval $(0,1)_R$ such that
$$\tag{$*$}\forall a\in A\,\exists b\in B\,\frac{a+b}2\in A\qquad\text{and}\qquad\forall b\in B\,\exists a\in A\,\frac{a+b}2\in B.$$
On the one hand, let $S\supseteq R$ be a rcf with the same Archimedean classes and $x\in S\smallsetminus R$. We can assume $x>1$. There exists $c\in R$ such that $c\sim x$; WLOG $c< x< 2c$. Then $0< x/c-1< 1$, and the Dedekind cut on $R$ determined by $x/c-1$ is easily seen to satisfy $(*)$.
On the other hand, assume the cut $\langle A,B\rangle$ is given. We define an ordering on the rational function field $F=R(x)$ as follows. Using the fact that every nonzero polynomial is a product of linear polynomials and polynomials of the form $(x-a)^2+b$, where $b>0$, we see that for every $f(x)/g(x)\in F$, there are $a\in A$ and $b\in B$ such that $f$ and $g$ have constant sign on $(a,b)_R$; we define the sign of $f(x)/g(x)$ to be the sign it assumes on $(a,b)_R$. This makes $F$ an ordered field. Let $S$ be its real closure. For a given $\alpha\in S$, there exists $c\in R$ such that $\alpha\sim c$ whenever:


*

*$\alpha=x-a$, $a\in R$. This follows from $(*)$.

*$\alpha=(x-a)^2+b$, $a,b\in R$, $b>0$. This follows from 1: if $u\sim u'$, $v\sim v'$, and $u,v>0$, then $u+v\sim u'+v'$.

*$\alpha\in F$. Every such $\alpha$ is a product of an element of $R$ and elements of the form 1 or 2 or their inverses, and $u\sim u'$ and $v\sim v'$ imply $uv\sim u'v'$ and $u/v\sim u'/v'$.

*$\alpha\in S$ is such that $\alpha^k\in F$ for some integer $k>0$. We have $\alpha^k\sim c$ by 3, hence $\alpha\sim\sqrt[k]c\in R$.

*$\alpha\in S$. We have $\sum_{i\le d}u\_i\alpha^i=0$ for some $u\_i\in F$, $u\_d\ne0$. Let $i$ be such that the Archimedean class of $u\_i\alpha^i$ is maximal. Since the sum above is $0$, there exists $j\ne i$ such that $u\_j\alpha^j\sim-u\_i\alpha^i$. Then $\alpha^{j-i}\sim-u\_i/u\_j$, hence $\alpha\sim c$ for some $c\in R$ by 4.
Thus $S$ is a proper rcf extension of $R$ with the same Archimedean classes as $R$.
A: Given, as Emil has noted, that every embedding of a real-closed ordered field into a real-closed ordered field is elementary, the following classical results from the theory of ordered algebraic systems provide an instructive alternative to Emil's nice answer to the question.The second of the two results is also a vast generalization of a result (essentially due to Hilbert 1900) about $\mathbb{R}$ intimated by Andreas.
Let $A$ be a real-closed ordered field, $G$ be an isomorphic copy of $A$'s divisible ordered abelian group of Archimedean classes and $\mathbb{R}(t^G)$ be the Hahn field with coefficients in $\mathbb{R}$ and exponents in $G$. 
I. The ordered abelian group of Archimedean classes of $\mathbb{R}(t^G)$ is isomorphic to $G$.
II. $\mathbb{R}(t^G)$ is real-closed. 
III. $\mathbb{R}(t^G)$ is a proper Archimedean extension of an isomorphic copy of $A$ if and only if $A$ is not isomorphic to $\mathbb{R}(t^G)$. 
I is trivial, II is back to W. Krull (1932) and III, which has a complicated history, is one of the familiar formulations of the ordered field version of Hahn's Embedding and Completeness Theorems for ordered abelian group and has its roots in the work of I. Kaplansky (1942) and H. Hahn (1907).
For an in-depth discussion of these and related results with complete references and historical details, see our 
Hahn’s Über die nichtarchimedischen Grössensysteme and the 
Development of the Modern Theory of Magnitudes and Numbers to Measure Them in From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics, edited by Jaakko Hintikka, Kluwer Academic Publishers, 1995, pp. 165-213.http://www.ohio.edu/people/ehrlich/
