Field Theory - Nicholson - Section 6.2 - Exercise 31

In Section 6.2 of Nicholson: Introduction to Abstract Algebra, Exercise 31 reads as follows:

Let be fields and let be transcendental over F.

(a) Show that

(b) Show that where F(x) is the field of quotients of the integral domain F[x].

(c) Show that every element , is transcendental over F.

Can someone help me approach this problem.

Peter

Re: Field Theory - Nicholson - Section 6.2 - Exercise 31

Since is a field extension of , there exists a function such that . Let . Let . Let for any . Essentially, what I am trying to do is find all elements of the form a+bu. I would think this is the way to start.

Re: Field Theory - Nicholson - Section 6.2 - Exercise 31

Thanks SlipEternal

One quick question:

You write: "Since E is a field extension of F, there exists a function such that ."

But is transcendental over F, so doesn't this mean that there is no polynomial in F[x] such that .

Can you clarify?

Thanks.

Peter

Re: Field Theory - Nicholson - Section 6.2 - Exercise 31

Right, I am getting tired. This should be much easier than that. So for any . Now, . Now through products of functions, you should be able to get any element of