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

Let L:K be a field extension.

is algebraic if and only if

where f is the minimal polynomial of alpha in K.

Now before I even get to the

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

are invertible) is fine except the proof I've seen states 'it is enough to show that the inverse of

is in the ring'. I don't understand why this is enough? Generally

is not of the form

with k,t in K (right?) so how is showing

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..