Don't understand proof of uniqueness theorem for polynom factorization

[swampwiz]

I don't understand proof of uniqueness theorem for polynomial factorization, as described in Stewart's "Galois Theory", Theorem 3.16, p. 38.

"For any subfield K of C, factorization of polynomials over K into irreducible polynomials in unique up to constant factors and the order in which the factors are written."

"Suppose that f = f₁ ... f_r = g₁ ... g_s where f is a polynomial over K and f₁ ... f_r , g₁ ... g_s are irreducible polynomials over K. If all the f_i are constant then ... so are all the g_j are constant."

So far so good.

"Otherwise, we may assume that no f_i is constant by dividing out all the constant terms."

How can this assumption be made? What if some of the f_i are constant, and some are not constant?

There is more unclear text here. Does anyone have a link to better explanation of this?

 

[fresh_42]

If $f_i=\lambda_if_i'$ then $f=\lambda_if_1\ldots f_{i-1}f_i'f_{i+1}\ldots f_r = \lambda_i f\,' = g_1\ldots g_s$. Now we can write $f\,'=\lambda_i^{-1}g_1\ldots g_s$ and get an equation without the constant factor on the left. By repeating this ,procedure, we will end up with an equation $f^{'\ldots '}=g_1'\ldots g_s'$ where the polynomial on the left does not have any scalar factors beside $1$ and especially all constant $f_i$ are eliminated. We then proceed with the reduced polynomial equation.

There is more unclear text here. Does anyone have a link to better explanation of this?

Unlikely, since these are standard techniques in algebra to reduce the problem to what counts and get rid of disturbing but irrelevant side effects. It would be better if you tried to figure out by yourself

What if some of the fi are constant, and some are not constant?

since it it a direct instruction of what to do: What if? And what will I do in such a case?
Basically along the lines of what I have done to answer the question.

 

[swampwiz]

I still don't get what Stewart is trying to say.

I do understand this (the way rasmhop explains it), which seems to be similar to what Stewart is trying to say.

https://www.physicsforums.com/threads/uniqueness-of-the-roots-of-a-polynomial-equation.371673/
I would think that Stewart is trying to say that every unique polynomial has a unique factorization into irreducible factors for whatever field is allowable for coefficients of the factors, and hence the complete polynomial. Or perhaps I am reading to much into what this theorem is supposed to mean?

 

[fresh_42]

Since I do not have the book and you did not quote what he said, I cannot answer your question.

It looks as a proof to show that every polynomial has a unique factorization.

 

[swampwiz]

Since I do not have the book and you did not quote what he say, I cannot answer your question.

It looks as a proof to show that every polynomial has a unique factorization.

I understand the proof when the field is the complex numbers (i.e., that's what the link to the other Physics Forums thread shows), but I would imagine that this is not the case when the proof is a field that is a subfield of the field of complex numbers. I could even see how any irreducible factor (i.e., irreducible in the subset) must necessarily be a product of the linear terms - i.e., ( x - r_j ) - such that this product is in the subfield. Since the end result is a set of factors in the subfield (including the possibility that this set is the original polynomial itself), at some point along the way of aggregation, there are resultant products within subfield of terms such that any factorization yields terms outside of the subfield, at which point, the end of the line of reducibility has been reached.

 

[swampwiz]

I've just thought of some other motivation for this theorem - does it have anything to do with the coefficients being prime numbers? Stewart goes on after this to discuss Gauss's Lemma, and the Eisenstein's Criterion right after.

 

[fresh_42]

I've just thought of some other motivation for this theorem

What theorem?

Since I do not have the book and you did not quote what he said, I cannot answer your question.

 

[swampwiz]

What theorem?

The theorem that Stewart references as THEOREM 3.16. He doesn't seem to give it a name.

 

[swampwiz]

I've found something that seems to cover what Stewart & fresh_42 are saying (@ p. 3):

https://math.okstate.edu/people/binegar/3613/3613-l19.pdf
EDIT: OK, I can follow the proof in these notes, so this question has been answered!