**[Edited so as to reflect answers and comments given so far]**

Let $F$ be a real closed field. Then by Artin-Schreier theory, $F[i]$ is algebraically closed. If $F$ is further assumed to have cardinality at most continnum, then by a classical theorem of Steinitz [stating the isomorphism of any two uncountable algebraically closed fields of the same cardinality and characteristic], we can conclude that $F[i]$ is isomorphic to a subfield of $\Bbb{C}$ of complex numbers.

In particular, if $F$ is a *non-archimedean* real closed field of cardinality continuum, then $F[i]$ is isomorphic to $\Bbb{C}$ [The proof uses the axiom of choice in a serious way, by the way].

We can therefore conclude:

**Theorem.** *Every nonarchimedean real closed field of power at most continuum is isomorphic to a subfield of $\Bbb{C}$.*

As a special case, we may conclude that there is a subfield $F$ of $\Bbb{C}$ such that $F$ is a non-archimdean real closed field that, furthermore, has a subfield isomorphic to the field $\Bbb{R}$ of real numbers.

The above considerations allow me to state my questions.

**Questions.**

**(a)** [UNANSWERED] *Is there an uncountable Borel non-Archimedean real closed field $F$ of $\Bbb{C}$?*

**NOTE**: In his comment below Dave Marker asks whether this question has a negative answer if we further assume that $F[i]=\Bbb{C}$, then Gerald Edgar pointed out in his comment that this is indeed the case; based on a result that appears in a joint paper of his with Chris Miller.

**(b)** [ANSWERED] Is it possible for an uncountable such $F$ to be at least **Lebesgue measurable** ?*

**NOTE**. (b) has been answered. First Martin Goldstern pointed out that (b) follows from $MA + \lnot CH$; and that (b) is also true in any universe of set theory obtained by adding a Cohen real. Then Gerald Edgar pointed out that (b) is provable outright in $ZFC$ [see the answers below].

**(c)** [UNANSWERED] If the answer to (a) is positive, does the answer change if we insist for $F$ to have a subfield isomorphic to $\Bbb{R}$.

`$F+iF$`

also has Hausdorff dimension $0$. ams.org/journal-getitem?pii=S0002-9939-02-06653-4 – Gerald Edgar May 28 '11 at 20:29