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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) = a_{n} x^{n} + a_{n-1} x^{n-1} + · · · + a_{1} x + a_{0} in D[x]
is called primitive if there is no irreducible element p in D such that p | a_{i} 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[x_{1} , x_{2} , ... , x_{n}]
in n indeterminates is a unique factorization domain.
Example 9.2.1.
The ring Z [
-5 ] is not
a unique factorization domain.