# Prove that an element is algebraic over a field extension

1. Jan 29, 2012

### TopCat

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 $K(u)(v) \cong K(u)[x]/(f)$ 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?

2. Jan 29, 2012

### morphism

Hint: K(u,v) is finite over K(u).

3. Jan 29, 2012

### TopCat

Thanks for the help, morphism.

So then I have [K(u,v):K] = [K(u,v):K(v)][K(v):K] = [K(u,v):K(u)][K(u):K] = n[K(u):K]. Since v trans|K, [K(v):K] is inf, and by equality [K(u):K] must be inf. But this means that u is trans|K, so there is an isomorphism from K(u) → K(v) mapping u → v. But this implies u is alg|K(v). Is that right, or did I get off track somewhere?

4. Jan 29, 2012

### morphism

My only objection is to the sentence "But this implies u is alg|K(v)."

How did you arrive at this conclusion?