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- Jun 22, 2012

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**Field Theory - Element u transcendental over F**

In Section 10.2 Algebraic Extensions in Papantonopoulou: Algebra - Pure and Applied, Proposition 10.2.2 on page 309 (see attachment) reads as follows:

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10.2.2 Proposition

Let E be a field, [TEX] F \subseteq E [/TEX] a subfield of E, and [TEX] \alpha \in E [/TEX] an element of E.

In E let

[TEX] F[ \alpha ] = \{ f( \alpha ) \ | \ f(x) \in F[x] \} [/TEX]

[TEX] F ( \alpha ) = \{ f ( \alpha ) / g ( \alpha ) \ | \ f(x), g(x) \in F[x] \ , \ g( \alpha ) \ne 0 \} [/TEX]

Then

(1) [TEX] F[ \alpha ] [/TEX] is a subring of E containing F and [TEX] \alpha [/TEX]

(2) [TEX] F[ \alpha ] [/TEX] is the smallest such subring of E

(3) [TEX] F( \alpha ) [/TEX] is a subfield of E containing F and [TEX] \alpha [/TEX]

(4) [TEX] F( \alpha ) [/TEX] is the smallest such subfield of E

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Papantonopoulou proves (1) and (2) (see attachment) and then writes:

" ... ... (3) and (4) are immediate from (1) and (2) since [TEX] F[ \alpha ] \subseteq E [/TEX] and E is a field, [TEX] F[ \alpha ] [/TEX] is an integral domain, and [TEX] F( \alpha ) [/TEX] is simply the field of quotients of [TEX] F[ \alpha ] [/TEX]. "

[Note: I do not actually follow this statement - can someone help clarify this "immediate" proof]

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**However ...**... in Nicholson: Introduction to Abstract Algebra, Section 6.2 Algebraic Extensions, page 279 (see attachment) we read:

" ... ... If u is transcendental over , it is routine to verify that

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

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

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*****My problem**with the above is that Papantonopoulou and Nicholson both give the same expression for [TEX] F( \alpha ) [/TEX] but Nicholson implies that the relation [TEX] F(u) = \{ f(u){g(u)}^{-1} \ | \ f(x), g(x) in F[x] \ ; \ g(x) \ne 0 \} [/TEX] is only the case

*???*

**if u is transcendental**Can someone please clarify this issue for me.

Peter

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