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# oct21 - d f divides h in Q x Since d f is primitive we...

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Math 5125 Friday, October 21 October 21, Ungraded Homework Exercise 9.3.2 on page 306 Prove that if f ( x ) and g ( x ) are polynomials with rational coef- ficients whose product f ( x ) g ( x ) has integer coefficients, then the product of any coefficient of g ( x ) with any coefficient of f ( x ) is an integer. Let h ( x ) = f ( x ) g ( x ) . We are given h ( x ) Z [ x ] and f , g Q [ x ] . Choose d Q such that d f is a primitive polynomial in Z [ x ] . Then we have h = ( d f )( g / d ) , so
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Unformatted text preview: d f divides h in Q [ x ] . Since d f is primitive, we deduce that d f divides h in Z [ x ] , equivalently g / d ∈ Z [ x ] . Therefore the product of any coefﬁcient of d f with any coefﬁcient of g / d is an integer. We conclude that the product of any coefﬁcient of f with any coefﬁcient of g is an integer, as required....
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