The Ring extension isomorphic to the field extension

Question

Let $\alpha$ be algebraic over $F$, with $F(\alpha)$ the smallest field containing both $F$ and $\alpha$, and with $F[\alpha]$ the smallest ring containing both $F$ and $\alpha$. I want to show $F(\alpha)=F[\alpha]$ which doesn't hold when $\alpha$ is transcendental (I think). The ring is clearly contained in the field, but given some $\sum a_k\alpha_k\in F(\alpha)$, I have no idea how to show it also exists in $F[\alpha]$.

Answer 1

Hint: All you have to prove is that, if α is algebraic over $F$, the $F[\alpha]$ is a field. Let $x=p(\alpha)\ne 0$. Consider multiplication by $x$ on $F[\alpha]$ and note 1) it is an $F$-linear map, 2) $F[\alpha]$ is a finite-dimensional $F$-vector space.