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

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

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

Thus $\displaystyle r \in R $

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

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

Question (2)

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

(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]Why does r and s being units allow us to conclude that $\displaystyle a(x) \in R[x] $?

I would be very appreciative of some help.

Peter