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Homework Statement
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).
Homework 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.
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?