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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, F \subseteq E a subfield of E, and \alpha \in E an element of E.
In E let
F[ \alpha ] = \{ f( \alpha ) \ | \ f(x) \in F[x] \}
F ( \alpha ) = \{ f ( \alpha ) / g ( \alpha ) \ | \ f(x), g(x) \in F[x] \ , \ g( \alpha ) \ne 0 \}
Then
(1) F[ \alpha ] is a subring of E containing F and \alpha
(2) F[ \alpha ] is the smallest such subring of E
(3) F( \alpha ) is a subfield of E containing F and \alpha
(4) F( \alpha ) 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 F[ \alpha ] \subseteq E and E is a field, F[ \alpha ] is an integral domain, and F( \alpha ) is simply the field of quotients of F[ \alpha ]. "
[Note: I do not actually follow this statement - can someone help clarify this "immediate" proof]================================================================================================
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
F(u) = \{ f(u){g(u)}^{-1} \ | \ f(x), g(x) in F[x] \ ; \ g(x) \ne 0
Hence F(u) \cong F(x) where F(x) is the field of quotients of the integral domain F[x]. ... ... "
=================================================================================================
***My problem with the above is that Papantonopoulou and Nicholson both give the same expression for F( \alpha ) but Nicholson implies that the relation F(u) = \{ f(u){g(u)}^{-1} \ | \ f(x), g(x) in F[x] \ ; \ g(x) \ne 0 \} is only the case if u is transcendental?
Can someone please clarify this issue for me.
Peter
In Section 10.2 Algebraic Extensions in Papantonopoulou: Algebra - Pure and Applied, Proposition 10.2.2 on page 309 (see attachment) reads as follows:
-----------------------------------------------------------------------------------------------------------
10.2.2 Proposition
Let E be a field, F \subseteq E a subfield of E, and \alpha \in E an element of E.
In E let
F[ \alpha ] = \{ f( \alpha ) \ | \ f(x) \in F[x] \}
F ( \alpha ) = \{ f ( \alpha ) / g ( \alpha ) \ | \ f(x), g(x) \in F[x] \ , \ g( \alpha ) \ne 0 \}
Then
(1) F[ \alpha ] is a subring of E containing F and \alpha
(2) F[ \alpha ] is the smallest such subring of E
(3) F( \alpha ) is a subfield of E containing F and \alpha
(4) F( \alpha ) is the smallest such subfield of E
--------------------------------------------------------------------------------------------------------------------------
Papantonopoulou proves (1) and (2) (see attachment) and then writes:
" ... ... (3) and (4) are immediate from (1) and (2) since F[ \alpha ] \subseteq E and E is a field, F[ \alpha ] is an integral domain, and F( \alpha ) is simply the field of quotients of F[ \alpha ]. "
[Note: I do not actually follow this statement - can someone help clarify this "immediate" proof]================================================================================================
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
F(u) = \{ f(u){g(u)}^{-1} \ | \ f(x), g(x) in F[x] \ ; \ g(x) \ne 0
Hence F(u) \cong F(x) where F(x) is the field of quotients of the integral domain F[x]. ... ... "
=================================================================================================
***My problem with the above is that Papantonopoulou and Nicholson both give the same expression for F( \alpha ) but Nicholson implies that the relation F(u) = \{ f(u){g(u)}^{-1} \ | \ f(x), g(x) in F[x] \ ; \ g(x) \ne 0 \} is only the case if u is transcendental?
Can someone please clarify this issue for me.
Peter
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