College Algebra Exam Review 299

College Algebra Exam Review 299 - (a Let b and a;a s be...

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6.6. UNIQUE FACTORIZATION DOMAINS 309 Proof. This is what was shown in the proof of Theorem 6.5.19 . n Proposition 6.6.16. An integral domain R is a unique factorization do- main if, and only if, R has the following two properties: (a) R satisfies the ascending chain condition for principal ideals. (b) Every irreducible in R is prime. Proof. This follows from Lemma 6.6.4 and Lemmas 6.6.13 through 6.6.15 . n Example 6.6.17. The integral domain Z Œ p ± (see Example 6.5.13 ) sat- isfies the ascending chain condition for principal ideals. On the other hand, Z Œ p ± has irreducible elements that are not prime. You are asked to ver- ify these assertions in Exercise 6.6.6 . Example 6.6.18. The integral domain R D Z C x Q ŒxŁ (see Example 6.5.14 ) does not satisfy the ascending chain condition for principal ideals, so it is not a UFD. However, irreducibles in R are prime. You are asked to verify these assertions in Exercise 6.6.7 . Exercises 6.6 6.6.1. Let R be a unique factorization domain.
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Unformatted text preview: (a) Let b and a ;:::;a s be nonzero elements of R . For d 2 R , show that bd is a greatest common divisor of f ba 1 ;ba 2 ;:::;ba s g if, and only if, d is a greatest common divisor of f a 1 ;a 2 ;:::;a s g . (b) Let f.x/ 2 RŒxŁ and let f.x/ D bf 1 .x/ , where f 1 is primitive. Conclude that b is a greatest common divisor of the coefficients of f.x/ . 6.6.2. (a) Let R be a commutative ring with identity 1. Show that the poly-nomial rings RŒx 1 ;:::;x n ± 1 ;x n Ł and .RŒx 1 ;:::;x n ± 1 Ł/Œx n Ł can be identified. (b) Assuming Theorem 6.6.7 , show by induction that if K is a field, then, for all n , KŒx 1 ;:::;x n ± 1 ;x n Ł is a unique factorization do-main. 6.6.3. (The rational root test) Use Gauss’s lemma (or the idea of its proof) to show that if a polynomial a n x n C²²²C a 1 x C a 2 Z ŒxŁ has a rational...
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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.

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