Unique factorization is polynomial-closed

Property-theoretic statement
The property of being a unique factorization domain is a polynomial-closed commutative unital ring property.

Verbal statement
The polynomial ring over a unique factorization domain is a unique factorization domain.

Proof
The proof essentially follows from Gauss's lemma. Gauss's lemma states that if f and $$g$$ are polynomials over a commutative unital ring $$R$$, then the content of the polynomial $$fg$$ is the product ideal of the content of $$f$$ and the content of $$g$$. In particular, the product of two primitive polynomials (viz polynomials without content) is a primitive polynomial.

The idea behind proving unique factorization is as follows:


 * First, express the given polynomial as a product of a primitive polynomial and the content. This can be done from the fact that the base ring is a unique factorization domain.


 * Then, perform unique factorization over the field of fractions. That this can be done follows from the fact that the polynomial ring over a field is a unique factorization domain.


 * For each factor, choose the unique primitive polynomial representing that factor. This gives a factorization of the primitive part.


 * The overall factorization is the factorization of the primitive part, along with the factorization of the content.