Some Background and Motivation: In this question, it is shown that an integral domain $D$ such that $F \subset D \subset E$, $E$ and $F$ fields with $[E:F]$ finite, is itself a field. However, a significantly more general result holds and seems worthy, of independent address; hence,
$F \subset E \tag 1$
be fields with
$[E:F] < \infty; \tag 2$
if $R$ is a ring such that
$F \subset R \subset E, \tag 3$
show that $R$ is in fact a field.