Unique factorization is polynomial-closed

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This article gives the statement, and possibly proof, of a commutative unital ring property satisfying a commutative unital ring metaproperty
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Statement

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 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.