Ytukyg

Let $E$ be an ordered field and let $F$ be a real closed subfield. We say that $E$ is $F$-archimedean if for each $ein E$ there is $xin F$ such that $-xle ele x$.

Is it true that if $E$ is $F$-archimedean then every interval in $E$ contains an element in $F$? That is, is it true that for every $e<e'$ in $E$ there is an element $xin F$ such that $e<x<e'$?

This is known if $F$ is the field of real algebraic numbers (in which case $E$ is an ordered subfield of $mathbb{R}$), and it seems to me that it should have an easy proof in the general case. However I cannot find neither an easy proof nor a counterexample.

Let $F$ be any real-closed field of uncountable cofinality. That is, every countable subset of $F$ is bounded. One can make such a field in a process of $omega_1$-many field extensions; alternatively, the ultrapower of $mathbb{R}$ by nonprincipal ultrafilter on $omega$ also has uncountable cofinality.

Let $E=F^omega/mu$ be an ultrapower of $F$ by a nonprincipal ultrafilter $mu$ on $omega$. We may identify $F$ with its canonical copy in $E$ using equivalence classes of constant functions $xmapsto [c_x]_mu$.

Since every function from $omega$ to $F$ is bounded by a constant function, it follows that $E$ is $F$-archimedean. But $F$ is not dense in $E$, since there are no constant functions between $[text{id}]_mu$ and $[text{id}+1]_mu$, where $text{id}:nmapsto n$, viewing $omegasubset F$.

Let $E$ be the real closure of $mathbb{Q}(x, y) = (mathbb{Q}(x))(y)$, with order given by $x > mathbb{Q}$ and$y > mathbb{Q}(x)$. In other words, positivity on $mathbb{Q}(x, y)$ is determined first by degree in $y$, and then by degree in $x$. Then let $F$ be the real closure of $mathbb{Q}(y)$.

First, we prove that $E$ is $F$-archimedean. Let $e in E$. There is some $e' in mathbb{Q}(x, y)$ with $e' > e > -e'$. Then $e'$ has degree $n$ in $y$ for some $y$; let $f = y^{n + 1}$. By the order on $mathbb{Q}(x, y)$, we have $f > e' > e > -e' > -f$. Therefore $E$ is $F$-archimedean.

On the other hand, clearly, there is no element of $F$ between $x$ and $x + 1$.

This is similar to user44191's answer but I want to put the emphasis on the fact that there is no reason that the cofinality of $F$ in $E$ (which is what ou call [$E$ is $F$-archimedean]) imply the density of $F$ in $E$.

Indeed, if $F$ is any non-archimedean ordered field, then the field $F(t)$ can be equipped with an order where $t$ is positive infinite but smaller than any positive infinite element of $F$. Thus $F$ is cofinal in $F(t)$ but not dense in it since no element of $F$ is close to $t$. To do so, say that a fraction $frac{P(t)}{Q(t)}$ is positive if $P(t)$ and $Q(t)$ have the same sign, where the sign of a polynomial $R(t)$ is positive if $R(s)$ is positive for sufficiently large finite element $s$ of $F$.

This is a special case of filling a cut in an ordered field using a simple extension.