Out of the blue, I asked myself the following question:
Is an infinite field always isomorphic to the fraction field of an integral domain which is itself not a field?
Please note that the above setup avoid answering by a field is its own fraction field.
I think this question is natural as it is kind of a converse to the following:
Proposition. The rings which can be embedded into some field are exactly the integral domains.
As fraction fields have no characteristic properties that I am aware of, I am lost on how to tackle this problem. Any enlightenment will be greatly appreciated!
Edit 1. My question only involves infinite field as a finite integral domain is itself a field and the result failed to be true in this case. - pointed out by carmichael561 in the comments below.
Is an infinite field of prime characteristic isomorphic to a non-trivial fraction field?