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..
It is enough because clearly is an integral domain...