Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

## § 9.2 Unique Factorization Domains

Lemma 9.2.1. Let D be a unique factorization domain, and let p be an irreducible element of D. If a,b are in D and p|ab, then p|a or p|b.

Definition 9.2.2. Let D be a unique factorization domain. A nonconstant polynomial

f(x) = an xn + an-1 xn-1 + · · · + a1 x + a0     in D[x]

is called primitive if there is no irreducible element p in D such that p | ai for all i.

Lemma 9.2.3. The product of two primitive polynomials is primitive.

Lemma 9.2.4. Let Q be the quotient field of D, and let f(x) be a polynomial in Q[x]. Then f(x) can be written in the form f(x) = (a/b)f*(x), where f*(x) is a primitive element of D[x], a,b are in D, and a and b have no common irreducible divisors.
This expression is unique, up to units of D.

Lemma 9.2.5. Let D be a unique factorization domain, let Q be the quotient field of D, and let f(x) be a primitive polynomial in D[x]. Then f(x) is irreducible in D[x] if and only if f(x) is irreducible in Q[x].

Theorem 9.2.6. If D is a unique factorization domain, then so is the ring D[x] of polynomials with coefficients in D.

Corollary 9.2.7. For any field F, the ring of polynomials

F[x1 , x2 , ... , xn]

in n indeterminates is a unique factorization domain.

Example 9.2.1. The ring Z [ -5 ] is not a unique factorization domain.