Is every field the field of fractions of an integral domain which is not itself a field?


What about the field of real numbers?



Is every field the field of fractions of an integral domain?
Every field $F$ of characteristic zero or of prime characteristic
but not algebraic over its prime field
is the field of fractions of a proper subring of $F$.
But no algebraic extension of $\mathbb F\_p$ is, since its only subrings are fields.


If $F$ is not an algebraic extension of some $\mathbb F\_p$
then $F$ contains a subring $A$ isomorphic to $\mathbb Z$
or $\mathbb F\_p[X]$. Each of these rings $A$ has a nontrivial
valuation $v$. The valuation $v$ can be prolonged to $F$.
Its valuation ring is a proper subring of $F$ whose quotient field
is $F$.


