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 problemwith 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