Page 218 - DMTH403_ABSTRACT_ALGEBRA

Unit 21: Unique Factorization Domains

Lemma: The product of two primitive polynomials is primitive. Notes
Lemma: 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: 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 6: If D is a unique factorization domain, then so is the ring D[x] of polynomials with
coefficients in D.

Corollary: For any field F, the ring of polynomials
F[x , x , ... , x ]
n
2
1
in n indeterminates is a unique factorization domain.
5
For example, the ring Z [  ] is not a unique factorization domain.
Self Assessment

1. If R is a UFD and a  R, with a  0 and being a .................., then a can be written as a product
of finite number of irreducible elements.
(a) invertible (b) non-invertible
(c) external (d) infinite
2. Any euclidean domains is a PID, it is also a ..................

(a) integral domain (b) UFD
(c) SFD (d) Ideal
3. Let R be a UFD. Then R(x) is a ..................
(a) UFD (b) SFD

(c) PID (d) Special range domain
4. In a UFD an element is prime iff it is ..................
(a) reducible (b) finite
(c) irreducible (d) infinite

5. Any two elements in a .................. have g.c.d.
(a) SFD (b) PID
(c) UFD (d) Domain

21.2 Summary

In a PID every prime ideal is a maximal ideal.

The definition and examples of a unique factorisation domain (UFD).

Every PID is a UFD, but the converse is not true. Thus, Z, F’ and F[x] are UFDs, for any

field F.

LOVELY PROFESSIONAL UNIVERSITY 211