2 Attachment(s)

Field Theory - Element u transcendental of 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, a subfield of E, and an element of E.

In E let

Then

(1) is a subring of E containing F and

(2) is the smallest such subring of E

(3) is a subfield of E containing F and

(4) 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 and E is a field, is an integral domain, and is simply the field of quotients of . "

[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

Hence 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 but Nicholson implies that the relation is only the case **if u is transcendental**???

Can someone please clarify this issue for me.

Peter

Re: Field Theory - Element u transcendental of F

I don't see that implication. I see that Nicholson uses that definition specifically for when is transcendental, but I do not see any indication by him that it would not be true when is algebraic. The proof that it is true for when is algebraic should be identical to the one where it is transcendental, possibly even easier. Given any , can you show that ? If so, then you are done.