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.