College Algebra Exam Review 295

College Algebra Exam Review 295 - Fx . Call an element of...

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6.6. UNIQUE FACTORIZATION DOMAINS 305 The following implications hold: pR maximal H) pR prime p prime p irreducible Proof. This follows from Lemma 6.5.18 and Lemma 6.6.4 n Example 6.6.6. In an UFD, if p is irreducible, pR need not be maximal. We will show below that Z ŒxŁ is a UFD. The ideal x Z ŒxŁ in Z ŒxŁ is prime but not maximal, since Z ŒxŁ=x Z ŒxŁ Š Z is an integral domain, but not a field. Polynomial rings over UFD’s The main result of this section is the following theorem: Theorem 6.6.7. If R is a unique factorization domain, then RŒxŁ is a unique factorization domain. It follows from this result and induction on the number of variables that polynomial rings KŒx 1 ; ±±± ;x n Ł over a field K have unique factoriza- tion; see Exercise 6.6.2 . Likewise, Z Œx 1 ; ±±± ;x n Ł is a unique factorization domain, since Z is a UFD. Let R be a unique factorization domain and let F denote the field of fractions of R . The key to showing that RŒxŁ is a unique factorization domain is to compare factorizations in RŒxŁ with factorizations in the Eu- clidean domain
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Unformatted text preview: Fx . Call an element of Rx primitive if its coefcients are relatively prime. Any element g.x/ 2 Rx can be written as g.x/ D dg 1 .x/; (6.6.1) where d 2 R and g 1 .x/ is primitive. Moreover, this decomposition is unique up to units of R . In fact, let d be a greatest common divisor of the (nonzero) coefcients of g , and let g 1 .x/ D .1=d/g.x/ . Then g 1 .x/ is primitive and g.x/ D dg 1 .x/ . Conversely, if g.x/ D dg 1 .x/ , where d 2 R and g 1 .x/ is primitive, then d is a greatest common divisor of the coefcients of g.x/ , by Exercise 6.6.1 . Since the greatest common divisor is unique up to units in R , it follows that the decomposition is also unique up to units in R . We can extend this discussion to elements of Fx as follows. Any element '.x/ 2 Fx can be written as '.x/ D .1=b/g.x/ , where b is a nonzero element of R and g.x/ 2 Rx . For example, just take b to be the...
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