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.
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.
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)
Then ... ... ... (b)
In equation (b) r is the coefficient of and coefficients of polynomials in R[x] must belong to R
Is this reasoning correct? Can someone please indicate any errors or confirm the correctness.
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 ? (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.