Polynomial Rings and UFDs - Dummit and Foote pages 303-304

In summary, the conversation discusses the proof of Theorem 7 in Dummit and Foote's Section 9.3, which states that R is a Unique Factorization Domain if and only if R[x] is a Unique Factorization Domain. The proof begins by stating that R[x] being a UFD implies that R is also a UFD. The question posed is where and how Dummit and Foote indicate or explain this statement, as it is not clear from the provided material. The suggested answer is that since the constants in R[x] are isomorphic to R, the unique factorization of polynomials in R[x] implies unique factorization of constants in R, thus making R a UFD.
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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 am so far unable to find anything that indicates to me that this is true.

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

Would appreciate some help.

Peter

[This question is also posted on MHF]
 
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Hello

I'm sorry if this is wrong. I haven't seen polynomial rings in at least 8 years. Anyway, I think that the answer is: If R[x] is a UFD, the polynomials that are constants, in particular, have unique factorization. Since the constants of R[x] are isomorphic to R, R is a UFD.

Does this make sense?
 

FAQ: Polynomial Rings and UFDs - Dummit and Foote pages 303-304

What is a polynomial ring?

A polynomial ring is a mathematical structure that consists of polynomials with coefficients from a given ring. The elements of a polynomial ring are formal expressions of the form a0 + a1x + a2x2 + ... + anxn, where ai are coefficients from the given ring and x is an indeterminate.

What is a UFD?

A UFD (unique factorization domain) is a type of ring in which every non-zero element can be uniquely factored into a product of irreducible elements. This property is similar to the fundamental theorem of arithmetic for integers, where every non-zero integer can be uniquely factored into primes.

How do you determine if a polynomial ring is a UFD?

In order for a polynomial ring to be a UFD, the ring must satisfy two conditions: it must be an integral domain (meaning it has no zero divisors) and every non-zero element must have a unique factorization into irreducible elements. To determine if a polynomial ring is a UFD, we can use the method of "taking out the common factors" to factorize a polynomial and then check if the factors are irreducible.

Can a polynomial ring have more than one UFD?

Yes, a polynomial ring can have multiple UFDs. This is because the property of being a UFD depends on the choice of coefficients and the indeterminate used in the polynomial ring. Different choices can lead to different factorizations, resulting in multiple UFDs for the same polynomial ring.

How are UFDs related to polynomial rings?

UFDs are important in polynomial rings because they allow us to factor polynomials in a unique way, which is useful in many areas of mathematics. In particular, the factorization of polynomials in a UFD can help us solve equations and prove theorems about polynomials. Additionally, polynomial rings can be used to construct UFDs by considering the ring of polynomials with coefficients in a UFD.

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