Unique factorization is polynomialclosed
From Commalg
This article gives the statement, and possibly proof, of a commutative unital ring property satisfying a commutative unital ring metaproperty
View all commutative unital ring metaproperty satisfactions  View all commutative unital ring metaproperty dissatisfactions Get help on looking up metaproperty (dis)satisfactions for commutative unital ring properties

Statement
Propertytheoretic statement
The property of being a unique factorization domain is a polynomialclosed 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 are polynomials over a commutative unital ring , then the content of the polynomial is the product ideal of the content of and the content of . 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.