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

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    Super Member Bernhard's Avatar
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    Field Theory - Nicholson - Section 6.2 - Exercise 31

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

    Let $\displaystyle E \supseteq F $ be fields and let $\displaystyle u \in E $ be transcendental over F.

    (a) Show that $\displaystyle F(u) = \{ f(u){g(u)}^{-1} \ | \ f,g \in F[x] ; g(x) \ne 0 \} $

    (b) Show that $\displaystyle F(u) \cong F(x) $ where F(x) is the field of quotients of the integral domain F[x].

    (c) Show that every element $\displaystyle w \in F(u), w \notin F $, is transcendental over F.

    Can someone help me approach this problem.

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

    Since $\displaystyle E$ is a field extension of $\displaystyle F$, there exists a function $\displaystyle f_0(x) \in F[x]$ such that $\displaystyle f_0(u) = 0$. Let $\displaystyle f_1(x) = f_0(x)+1$. Let $\displaystyle f_u(x) = x$. Let $\displaystyle f_k(x) = kf_1(x)$ for any $\displaystyle k \in F$. 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.
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    Super Member Bernhard's Avatar
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    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 $\displaystyle f_0(x) \in F[x] $ such that $\displaystyle f_0(u) = 0 $."

    But $\displaystyle u \in E $ is transcendental over F, so doesn't this mean that there is no polynomial $\displaystyle f_0(x) $ in F[x] such that $\displaystyle f_0(u) = 0 $.

    Can you clarify?

    Thanks.

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

    Right, I am getting tired. This should be much easier than that. So $\displaystyle f_{a+bu}(x) = a+bx$ for any $\displaystyle a,b \in F$. Now, $\displaystyle f_{a+bu}(u) = a+bu$. Now through products of functions, you should be able to get any element of $\displaystyle F(u)$
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