TITLE: Integral domain with fraction field equal to $\mathbb{R}$
QUESTION [13 upvotes]: I wonder if there is an integral domain $A\subseteq \mathbb{R}$ which is not a field, and such that the field of fractions of $A$ is equal to $\mathbb{R}$?
Edit: here as a possible direction: it is known that there is no finite field extension $k\subseteq \mathbb{R}$. Given such an $A$, can we produce such a $k$, contradicting this theorem?

REPLY [6 votes]: The question was asked and answered on MathOverflow in 2010, in somewhat greater generality, and by two different methods.