- 9.1 Principal ideal domains
- 9.2 Unique factorization domains

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**(i)**(ab) (b) for all nonzero a,b in D, and**(ii)**for any nonzero elements a,b in D there exist q,r in D such that a = bq + r,

**9.1.2. Theorem.**
Any Euclidean domain is a principal ideal domain.

Let a and b be elements of a commutative ring R with 1.
Then a is called an
** associate**
of b if a = bu for some unit u in R.

**9.1.4. Definition.**
Let a and b be elements of a commutative ring R with identity.
An element d of R is called a
** greatest common divisor**
of a and b if

**(i)**d | a and d | b, and**(ii)**if c | a and c | b, for c in R, then c | d.

In the situation of the above proposition,
in an integral domain D,
we say that a and b are
** relatively prime**
if aD+bD=D.

**9.1.7. Definition.**
Let R be a commutative ring with identity.
A nonzero element p of R is said to be
** irreducible**
if

**(i)**p is not a unit of R, and**(ii)**if p = ab for a,b in R, then either a or b is a unit of R.

**9.1.9. Proposition.**
Let D be a principal ideal domain,
and let p be a nonzero element of D.
Then p is irreducible in D if and only if
pD is a prime ideal of D.

**9.1.10. Definition.**
Let D be an integral domain.
Then D is called a
**unique factorization domain**
if

**(i)**each nonzero element a of D that is not a unit can be expressed as a product of irreducible elements of D, and**(ii)**in any two such factorizations a = p_{1}p_{2}· · · p_{n}= q_{1}q_{2}· · · q_{m}the integers n and m are equal and it is possible to rearrange the factors so that q_{i}is an associate of p_{i}, for 1 i n.

I
_{1}
I
_{2}
I
_{3}
· · · ,

**9.1.12. Theorem.**
Any principal ideal domain is a unique factorization domain.

**9.2.2. Definition.**
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.

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

**9.2.4. Lemma.**
Let Q be the quotient field of D, and let
f(x) 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 D,
and a and b have no common irreducible divisors.
This expression is unique, up to units of D.

**9.2.5. Lemma.**
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].

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

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

F[x_{1}
, x_{2}
, ... , x_{n}]

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