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.
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
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.
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