Let $R$ be an integral domain and $F$ its field of quotients.
If $R$ is itself a field, show that $R = F$.
Intuitively this makes sense. This follows directly from the following theorem:
Let $F$ be the field of quotients of an integral domain $R$. If $K$ is a field containing $R$, then $K$ contains a subfield $E$ such that $R\subseteq E \subseteq K$ and $E \cong F$.
This theorem asserts that the field of quotients of an integral domain $R$ is the smallest field that contains $R$.
Therefore it makes sense that if $R$ is itself a field, then $R=F$, but how can I prove it directly?