UNIQUE FACTORIZATION

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

Chapter 9

9.1 Principal ideal domains
9.2 Unique factorization domains

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Principal ideal domains

9.1.1. Definition. An integral domain D is called a Euclidean domain if for each nonzero element x in D there is assigned a nonnegative integer (x) such that
(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,

where either r = 0 or (r) < (b).

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.

9.1.6. Proposition. Let D be a principal ideal domain. If a and b are nonzero elements of D, then D contains a greatest common divisor of a and b, of the form as+bt for s,t in D. Furthermore, any two greatest common divisors of a and b are associates.

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.

9.1.8. Proposition. Let p be an irreducible element of the principal ideal domain D. If a,b D and p|ab, then either p|a or p|b.

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 = p1 p2 · · · pn = q1 q2 · · · qm the integers n and m are equal and it is possible to rearrange the factors so that qi is an associate of pi, for 1 i n.

9.1.11. Lemma. Let D be a principal ideal domain. In any collection of ideals

I 1 I 2 I 3 · · · ,

there is a subscript m such that In = Im for all n>m.

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

Unique factorization domains

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

9.2.2. Definition. 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.

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[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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