# Thread: Polynomial Rings and UFDs - Dummit and Foote pages 303-304

1. ## Polynomial Rings and UFDs - Dummit and Foote pages 303-304

I am reading Dummit and Foote Section 9.3 Polynomial Rings That are Unique Factorization Domains (see attachment Section 9.3 pages 303 -304)

I am working through (beginning, anyway) the proof of Theorem 7 which states the following:

"R is a Unique Factorization Domain if and only if R[x] is a Unique Factorization Domain"

The proof begins with the statement:

Proof: We have indicated above the R[x] a Unique Factorization Domain forces R to be a Unique Factorization Domain"

Question! Can anyone explain to me how and where D&F indicate or explain this first statement of the proof. Secondly, what is the explanation

I have provided Section 9.3 up to Theorem 7 as an attachment.

Would appreciate some help.

Peter

2. ## Re: Polynomial Rings and UFDs - Dummit and Foote pages 303-304

It's easier to see the contrapositive: If $\displaystyle R$ is not a UFD, how can $\displaystyle R[x]$ possibly one? If you think of elements of R as constant polynomials, this should be trivial.