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.
- Fall '08