Unformatted text preview: 6.6. UNIQUE FACTORIZATION DOMAINS (b) (c) 303 Show that KŒŒx is a principal ideal domain. Hint: Let J be
an ideal of KŒŒx. Let n be theP
least integer such that J has
an element of the form ˛n x n C j >n ˛j x j . Show that J D
x n KŒŒx.
Show that KŒŒx has a unique maximal ideal M , and
KŒŒx=M Š K . 6.5.24. Fix a prime number p and consider the set Qp of rational numbers
a=b , where b is not divisible by p . (The notation Qp is not standard.)
Show that Qp is a principal ideal domain with a unique maximal ideal M .
Show that Qp =M Š Zp . 6.6. Unique Factorization Domains
In the ﬁrst part of this section, we discuss divisors in a unique factorization domain. We show that all unique factorization domains share some
of the familiar properties of principal ideal. In particular, greatest common
divisors exist, and irreducible elements are prime.
Lemma 6.6.1. Let R be a unique factorization domain, and let a 2 R be
a nonzero, nonunit element with irreducible factorization a D f1 fn .
If b is a nonunit factor of a, then there Q
exist a nonempty subset S of
f1; 2; : : : ; ng and a unit u such that b D u i 2S fi . Proof. Write a D bc . If c is a unit, then b D c 1 a D c 1 f1 fn ,
which has the required form. If c is not a unit, consider irreducible factorizations of b and c , b D g1 g` and c D g`C1 gm . Then a D
g1 g` g`C1 gm is an irreducible factorization of a. By uniqueness
of irreducible factorization, m D n, and the gi ’s agree with the fi ’s
up to order and multiplication by units. That is, there is a permutation
of f1; 2; : : : ; ng such that each gj is an associate of f .j / . Therefore
b D g1 : : : g` is an associate of f .1/ f .`/ .
I Lemma 6.6.2. In a unique factorization domain, any ﬁnite set of nonzero
elements has a greatest common divisor, which is unique up to multiplication by units. ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08