Let $D$ be an integral domain and let $F$ be the field of quotients of $D$. Show that if $E$ is any field that contains $D$, then, $E$ contains a sub field that is ring isomorphic to $F$. Hence, the field of quotients of an integral domain is the smallest field containing $D$.
Attempt: Let $S$ be a sub ring of the field of quotient $F$ such that $S \approx D$
We need to show that $F \approx $ a sub field of $E$.
Let $F'$ be the field of quotients of $F$.
If $K$ be a field, then, the field of quotients of $K$ is ring isomorphic to $K$ is a result.
Hence, We know that $F' \approx F$ . Hence, $F'$ must contain $D$.
Since, $F'$ is a field, $\implies F'$ is a sub field of $E$.
Hence Proved that there exists a sub field of $E$ which is isomorphic to $F$.
Is my attempt correct?
Please note that my book hasn't yet introduced polynomials, reducability, divisibility in integral domain or field extensions.
Thank you for your help..