College Algebra Exam Review 296

College Algebra Exam Review 296 - 306 6. RINGS product of...

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Unformatted text preview: 306 6. RINGS product of the denominators of the coefﬁcients of '.x/. Factoring g.x/ as above gives '.x/ D .d=b/f .x/; (6.6.2) where f .x/ is primitive in RŒx. This decomposition is unique up to units in R. In fact, if .d1 =b1 /f1 .x/ D .d2 =b2 /f2 .x/; where f1 and f2 are primitive in RŒx, then d1 b2 f1 .x/ D d2 b1 f2 .x/. By the uniqueness of the decomposition 6.6.1 for RŒx, there exists a unit u in R such that d1 b2 D ud2 b1 . Thus d1 =b1 D ud2 =b2 . Example 6.6.8. Take R D Z. 7=10 C 14=5x C 21=20x 3 D .7=20/.2 C 8x C 3x 3 /; where 2 C 8x C 3x 3 is primitive in ZŒx. Lemma 6.6.9. (Gauss’s lemma). Let R be a unique factorization domain with ﬁeld of fractions F . (a) The product of two primitive elements of RŒx is primitive. (b) Suppose f .x/ 2 RŒx. Then f .x/ has a factorization f .x/ D '.x/ .x/ in F Œx with deg.'/; deg. / 1 if, and only if, f .x/ has such a factorization in RŒx. P P Proof. Suppose that f .x/ D ai x i and g.x/ D bj x j are primitive in RŒx. Suppose p is irreducible in R. There is a ﬁrst index r such that p does not divide ar and a ﬁrst index s such thatP does not divide p r Cs in f .x/g.x/ is a b C bs rs i <r ai br Cs i C P. The coefﬁcient of x ar Cs j bj . By assumption, all the summands are divisible by p , j <s except for ar bs , which is not. So the coefﬁcient of x r Cs in fg.x/ is not divisible by p . It follows that f .x/g.x/ is also primitive. This proves part (a). Suppose that f .x/ has the factorization f .x/ D '.x/ .x/ in F Œx with deg.'/; deg. / 1. Write f .x/ D ef1 .x/, '.x/ D .a=b/'1 .x/ and .x/ D .c=d / 1 .x/, where f1 .x/, '1 .x/, and 1 .x/ are primitive in RŒx. Then f .x/ D ef1 .x/ D .ac=bd /'1 .x/ 1 .x/. By part (a), the product '1 .x/ 1 .x/ is primitive in RŒx. By the uniqueness of such decompositions, it follows that .ac=bd / D eu, where u is a unit in R, so f .x/ factors as f .x/ D ue'1 .x/ 1 .x/ in RŒx. I Corollary 6.6.10. If a polynomial in ZŒx has a proper factorization in QŒx, then it has a proper factorization in ZŒx. ...
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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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