If we take any integral domain, then we can define a field of fractions by taking equivalence classes of ordered pairs of elements, the same way that the rational numbers are constructed from the integers. My question is:
What fields (of characteristic $0$) are isomorphic to the field of fractions of some integral domain (that's not a field)?
For instance, is the field of constructible real numbers a nontrivial field of fractions? What about the algebraic real numbers? What about arbitrarily real closed fields? And what about if we restrict ourselves to integral domains which are models of Peano arithmetic, or models of Robinson arithmetic? (EDIT: for those less acquainted with logic and model theory, let me ask this: what if we restricted the integral domains to ones that are discretely ordered rings?) I should mention that my motivation for asking these sorts of questions is my MathOverflow question.
Any help would be greatly appreciated.
Thank You in Advance.
Answer
This is a partial answer (it answers your first question).
Every field of characteristic zero is the fraction field of some integral domain which is not a field. Indeed, let $k$ be your field and let $(X_i)$ be a transcendence basis for $k$ over $\mathbb{Q}$. Consider the ring $R$ which is the integral closure of $\mathbb{Z}[\{X_i\}]$ in $k$. Note then that $R\ne k$ (since integral extensions preserve dimension), but it's a common fact that $k=\text{Frac}(R)$.
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