1. The problem statement, all variables and given/known data Let F|K be a field extension. If v e F is algebraic over K(u) for some u e F and v is transcendental over K, then u is algebraic over K(v). 2. Relevant equations v transcendental over K implies K(v) iso to K(x). Know also that there exists f e K(u)[x] with f(v) = 0. 3. The attempt at a solution Want to show that there exists h e K(v)[x] with h(u) = 0. I'm trying to find this directly since I don't see a contrapositive proof working out. I feel like I should use v alg|K(u) to get that [itex]K(u)(v) \cong K(u)[x]/(f)[/itex] though I'm not sure how to get to that h in K(v)[x]. Somehow pass to that quotient field and show that u is a root of some remainder polynomial of f?