If R is an integral domain, show that the field of quotients Q is the smallest field containing R in the following sense: If R is a subset of F, where F is a field, show that F has subfield K such that R is a subset of K and K is isomorphic to Q.
I have trouble interpreting this question. My understanding is that we assume R is a subset of F, we want to prove that there exists K which is a subfield of F such that R is a subset of K and K is isomorphic to Q. That means we want to prove K is a subfield of F. Am I right? If I am right, then how to prove K is a subfield of F. Do I have to prove K is a subring first, then prove K is a field?