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oct19 - = f x g x We use the division algorithm there are...

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Math 5125 Wednesday, October 19 October 19, Ungraded Homework Exercise 9.2.1 on page 301 Let F be a field, let f ( x ) F [ x ] , and let bars denote passage to the quotient F [ x ] / ( f ( x )) . Prove that for each g ( x ) , there is a unique polynomial g 0 ( x ) of degree n - 1 (or g 0 = 0) such that g ( x ) = g 0 ( x ) . By definition, g ( x
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Unformatted text preview: ) = ( f ( x )) + g ( x ) . We use the division algorithm: there are unique poly-nomials q ( x ) and g ( x ) with either g = 0 or deg g < deg f such that g = q f + g . Then g ( x ) = g ( x ) and the result follows....
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