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Unformatted text preview: ( x ) = ( x âˆ’ a 1 ) Â·Â·Â· ( x âˆ’ a n ) âˆˆ R [ x ] , where R is a commutative ring. Show that f ( x ) has no repeated roots (that is, all the a i are distinct) if and only if the gcd ( f , f ) = 1, where f is the derivative of f . Solution. Exercise 3.37 says that x âˆ’ a is a common divisor of f ( x ) and f if and only if ( x âˆ’ a ) 2  f ( x ) ; that is, if and only if f ( x ) has repeated roots. 3.68 Let âˆ‚ be the degree function of a Euclidean ring R . If m , n âˆˆ N and m â‰¥ 1, prove that âˆ‚ is also a degree function on R , where âˆ‚ ( x ) = m âˆ‚( x ) + n for all x âˆˆ R . Conclude that a Euclidean ring may have no elements of degree 0 or degree 1. Solution. If x , y âˆˆ R are nonzero, then âˆ‚ ( x ) = m âˆ‚( x ) + n â‰¤ m âˆ‚( xy ) + n = âˆ‚ ( xy )....
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
 KeithCornell

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