MHB GCDs of Polynomials: Reading Rotman's Corollary 3.58

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I am reading Joseph J.Rotman's book, A First Course in Abstract Algebra.

I am currently focused on Section 3.5 From Polynomials to Numbers

I need help with the statement and meaning of Corollary 3.58

The relevant section of Rotman's text reads as follows:https://www.physicsforums.com/attachments/4547In the above text (in the statement of the Corollary) we read the following:

" ... ... (ii) Every two polynomials $$f(x)$$ and $$g(x)$$ have a unique gcd. ... ... "
My question (which some may regard as pedantic :) ) is as follows:

Does Rotman actually mean ...

" (ii) Every two polynomials $$f(x)$$ and $$g(x)$$ have a unique monic gcd. ... ... "

Can someone please confirm that my interpretation is correct?

Peter
 
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Peter said:
I am reading Joseph J.Rotman's book, A First Course in Abstract Algebra.

I am currently focused on Section 3.5 From Polynomials to Numbers

I need help with the statement and meaning of Corollary 3.58

The relevant section of Rotman's text reads as follows:In the above text (in the statement of the Corollary) we read the following:

" ... ... (ii) Every two polynomials $$f(x)$$ and $$g(x)$$ have a unique gcd. ... ... "
My question (which some may regard as pedantic :) ) is as follows:

Does Rotman actually mean ...

" (ii) Every two polynomials $$f(x)$$ and $$g(x)$$ have a unique monic gcd. ... ... "

Can someone please confirm that my interpretation is correct?

Peter
I think gcd is defined to a monic by default. Please check the definition in the book. If it doesn't say monic specifically then yes, you can have many gcds.
 

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