Are all fields the field of fractions of an integral domain that is not a field?

Question

A field $F$ is said to be fracfield if there exist a integral domain $R$ which is not a field such that fraction field of $R$ is $F$. Fields such as $\mathbb Q,$ field of rational functions, number fields etc are fracfields. I was not able to determine if $\mathbb R, \mathbb C$ are fracfields. Are there fields which are not frac fields? Are there a criteria for determining the same?

Answer 1

This question was raised in math.overflow in 2010 (but I cannot tag this as a duplicate of that). To prevent it from remaining unanswered, I am posting the answer posted there by Robin Chapman as a CW.

Every field of characteristic zero is the quotient field of a proper subring which is not a field. A field of characteristic $p\gt 0$ is the quotient field of a proper subring which is not a field if and only if it is not an algebraic extension of its prime field.

For fields of characteristic zero, we know they contain a subring isomorphic to $\mathbb{Z}$; fields of characteristic $p\gt 0$ that are not algebraic over their prime field contain a subring isomorphic to the polynomial ring $\mathbf{F}_p[x]$. Either one of them has nontrivial valuations, which can be extended to the entire field. Because it is a nontrivial valuation, the valuation ring is a proper subring of the field which is not a field, and its field of fractions is the whole field.

On the other hand, an algebraic extension of $\mathbf{F}_p$ has the property that all its subrings are fields, so such a field is not the field of fractions of a subring that is not a field.