Unformatted text preview: FŒxŁ . Call an element of RŒxŁ primitive if its coefﬁcients are relatively prime. Any element g.x/ 2 RŒxŁ 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) coefﬁcients 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 coefﬁcients 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 FŒxŁ as follows. Any element '.x/ 2 FŒxŁ can be written as '.x/ D .1=b/g.x/ , where b is a nonzero element of R and g.x/ 2 RŒxŁ . For example, just take b to be the...
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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