# Math Help - Polynomial Rings and UFDs

1. ## Polynomial Rings and UFDs

Exercise 1, Section 9.3 in Dummit and Foote, Abstract Algebra, reads as follows:

Let  be an integral domain with quotient field  and let  be monic. Suppose  factors nontrivially as a product of monic polynomials in , say , and that . Prove that  is not a unique factorization domain. Deduce that  is not a unique factorization domain.
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I found a proof on Project Crazy Project which reads as follows:

Suppose to the contrary that  is a unique factorization domain. By Gauss’ Lemma, there exist  such that  and . Since , , and  are monic, comparing leading terms we see that . Moreover, since  is monic and , we have . Similarly, , and thus  is a unit. But then , a contradiction. So  cannot be a unique factorization domain.

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Question (1)

Consider the following statement in the Project Crazy Project Proof:

"Moreover, since  is monic and , we have ."

My reasoning regarding this statement is as follows:

Consider $a(x) = x^n + a_{n-1}x^{n-1} + ... ... + a_1x + a_0$ ... ... ... (a)

Then $ra(x) = rx^n + ra_{n-1}x^{n-1} + ... ... + ra_1x + ra_0$ ... ... ... (b)

In equation (b) r is the coefficient of $x^n$ and coefficients of polynomials in R[x] must belong to R

Thus $r \in R$

Is this reasoning correct? Can someone please indicate any errors or confirm the correctness.

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Question (2)

The last part of the Project Crazy Project (PCP) reads as follows:

Moreover, since  is monic and , we have . Similarly, , and thus  is a unit. But then , a contradiction. So  cannot be a unique factorization domain.

Why does r and s being units allow us to conclude that $a(x) \in R[x]$? (I am assuming that the author of PCP has made an error in writing R in this expression and that he should have written R[x]

I would be very appreciative of some help.

Peter