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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.
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