Problem: If F is algebraic over K and D is an integral domain such that $K\subset D\subset F$, then D is a field.

Sketch of Proof: We want to show every element of D is a unit. Let u in D. Since F is a field, u^{-1} exists at least in F. Since F is algebraic over K, there exists f in K[x]\{0} such that f(u^{-1})=b_0+b_1u^{-1}+...+b_m u^{-m}=0. Now using a nonzero element of D (such as u^{m-1}) I want to be able to factor the last expression and divide by the nonzero element to obtain a polynomial expression for u^{-1} in K[u], which will complete the proof since K[u]\subset D.

Arlington