GCDs of Polynomials: Reading Rotman's Corollary 3.58

[Math Amateur]

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 \(\displaystyle f(x)\) and \(\displaystyle g(x)\) have a unique gcd. ... ... "
My question (which some may regard as pedantic :) ) is as follows:

Does Rotman actually mean ...

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

Can someone please confirm that my interpretation is correct?

Peter

 

[caffeinemachine]

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 \(\displaystyle f(x)\) and \(\displaystyle g(x)\) have a unique gcd. ... ... "
My question (which some may regard as pedantic :) ) is as follows:

Does Rotman actually mean ...

" (ii) Every two polynomials \(\displaystyle f(x)\) and \(\displaystyle g(x)\) 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.