ABSTRACT
After having discussed the general theory of factorizations in integral domains, we now want to apply this theory to polynomial rings. The only class of examples of unique factorization domains we know as yet is the class of principal ideal domains. This class comprises the polynomial rings K[x] where K is a field because these rings are Euclidean. On the other hand, it is true that ℤ [ x ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/b1e97aa6-f63a-4d22-960d-a98dde0e94f9/content/eq1611.tif"/> is a unique factorization domain even though it is not a principal ideal domain. (For example, the ideal 〈2, x〉 is not a principal ideal.) The fact that ℤ [ x ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/b1e97aa6-f63a-4d22-960d-a98dde0e94f9/content/eq1612.tif"/> is in fact a unique factorization domain is contained in a general result which we are going to establish now, namely, that if R is a unique factorization domain then so is the polynomial ring R[x]. For the proof of this statement, it is convenient to introduce the following terminology.
