Is u Algebraic over Extension Field E?

[Driessen12]

Homework Statement

if F=k(u), where u is transcendental over the field k. If E is a field such that E is an extension of K and F is an extension of E, then show that u is algebraic over E

Homework Equations

The Attempt at a Solution

i''m having trouble starting this proof, any ideas? any help would be appreciated

 

[VKint]

Well, if $$E$$ is a field strictly bigger than $$k$$, then it must contain some $$\alpha$$ not in $$k$$. By assumption, $$\alpha \in F$$. As a vector space, $$F$$ is infinite-dimensional over $$k$$, but it does have a very convenient basis. Expand $$\alpha$$ in this basis (noting that in this context "basis" means "Hamel basis," so infinite linear combinations are not allowed). This should give you an interesting relation between $$u$$ and $$\alpha$$. Do you see why this solves the problem?

 

[Driessen12]

I think I must prove this using the idea of irreducible(minimal) polynomials, and algebraic elements

 

[Driessen12]

but I think the interesting connection is that they are linearly independent and span E, making E contain u?

 

[VKint]

I think I must prove this using the idea of irreducible(minimal) polynomials, and algebraic elements

The problem is much more straightforward than that. Don't think too hard. :P

Actually, $$E$$ doesn't have to contain $$u$$. If it did, then it would necessarily also contain $$F = k(u)$$ (since $$k(u)$$ is the smallest field containing both $$k$$ and $$u$$). However, it does have to contain something which is not in $$k$$, and that something can be expanded in powers of $$u$$. Think a little about this, and you'll realize it's exactly what you want.