Transcendental extension fields

Hi, I'm having problems with the second part of this problem.

Let be an extension of and transcendental over .

Let with . Show:

a) is transcendental over .

b) If and , then is transcendental over .

For part a, I assumed otherwise, so there exists a such that , but and which contradicts being transcendental over .

For part b, I'm not sure. I tried doing it by contradiction also, but I'm not getting anywhere. I feel like it's relatively simple, and I'm just forgetting something obvious. Any help would be appreciated! Thanks!

Re: Transcendental extension fields

Tell me if you have covered such a result or not but finite sums and products of algebraic elements are algebraic. So we can prove the contrapositive. Let be algebraic. We need to show . To this end, is just a finite sum of products of and the coefficients of , that is is algebraic and cannot be transcendental, i.e. .

Re: Transcendental extension fields

Another way. If is algebraic over , then is finite and as a consequence is an algebraic extension of As , would be algebraic over (contradiction).

Re: Transcendental extension fields

I have not encountered that result, although it seems to make sense (it would be frightening to think of transcendentals being the result of finite sums and/or products of algebraic elements).

I like Fernando's proof though, and it's something I should have realized, it's just that the problem got my mind set on using functions.

This was my solution:

(Preliminary stuff)

Notation: and

If then

Proof: Since , . Now, but , thus .

Similarly, If then .

Thus, if , then and .

Let , so .

Now .

Note that: , (because if it is in then which means which contradicts being transcendental).

So, using the preliminary stuff, implies that and since , we have that .

So, and . Thus, by part (a), is transcendental.

Is there anything wrong with what I've done?

What I think I basically did is:

If such that , then such that .