This is an exercise in some textbooks.
Let $E$ be an algebraic extension of $F$. Suppose $R$ is ring that contains $F$ and is contained in $E$. Prove that $R$ is a field.
The trouble is really with the inverse of $r$, where $r\in R$. How to prove that $r^{-1}\in R$, in apparent lack of a characterization of $R$.
It occurred to me to use the smallest field containing $R$ ($R$ is easily shown to be an integral domain), that's the field of quotients, and proving that it's $R$ itself, but I don't really know how to proceed.
A not-too-weak, not-too-strong hint will be much appreciated.
Beware $ $ Readers seeking only hints should beware that there is now a complete answer.
Best Answer
Well, you know that $R$ is contained in an algebraic extension of $F$, so you should use it somehow. It directly implies that there is a polynomial (of minimal degree, say) with coefficients in $F$ which annihilates $r$. Can you manufacture an inverse for $r$ using this polynomial?