Field of Fractions and Extensions

Hello, I'd like some help regarding field extensions.

Let L:K be a field extension. $\displaystyle \alpha \in L$ is algebraic if and only if $\displaystyle K[\alpha] = K(\alpha) \simeq F[X]/(f) $ where f is the minimal polynomial of alpha in K.

Now before I even get to the $\displaystyle F[X]/(f)$ I'd like some clarification about the equality between the polynomials in alpha and the rational functions in alpha.

I know one inclusion is trivial; the other inclusion (showing that all non-zero elements in $\displaystyle K[\alpha]$ are invertible) is fine except the proof I've seen states 'it is enough to show that the inverse of $\displaystyle \alpha$ is in the ring'. I don't understand why this is enough? Generally $\displaystyle K[\alpha]$ is not of the form $\displaystyle k+t\alpha$ with k,t in K (right?) so how is showing $\displaystyle \alpha ^{-1}$ is in the ring enough to show all the other inverses are in the ring?

For the isomorphism I'm not sure what map to use..