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Field Theory - Nicholson - Section 6.2 - Exercise 31

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
In Section 6.2 of Nicholson: Introduction to Abstract Algebra, Exercise 31 reads as follows:

Let [TEX] E \supseteq F [/TEX] be fields and let [TEX] u \in E [/TEX] be transcendental over F.

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

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

(c) Show that every element [TEX] w \in F(u), w \notin F [/TEX], is transcendental over F.

Can someone help me approach this problem.

Peter

[This has also been posted on MHF]