Given a field extension $K/F$, $\alpha \in K$, why is it the case that $F(\alpha)$, i.e. the subfield of K generated by $\alpha$, is isomorphic to the fraction field of $F[\alpha]$?
My understanding is that as $F[\alpha]$ is the smallest subring that contains $\alpha$, and to make it into a field, we take the ratios. And somehow, this makes it the smallest subfield of $K$ that contains $\alpha$. I can't construct a proper proof for this and is driving me crazy.
Any help would be appreciated!
Cheers