TITLE: If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field.
QUESTION [17 upvotes]: Possible Duplicate:
Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field 

I need to prove this result, but the only starting point I think of is to argue by contradiction, by assuming $x$ is non-invertible and extending it to a basis over $K$ but have no idea how to go further than that. Any ideas?

REPLY [7 votes]: Hint $\ $ Being finite dimensional, the extension is algebraic. But a domain algebraic over a field is a field, since one obtains an inverse of an element $\ne 0$ from a minimal polynomial. See my post here for much further discussion of this and related methods of lifting the existence of inverses (e.g. the well-known method of rationalizing denominators).