Unformatted text preview: largest degree ) of A ( x ) and B ( x ) over some ﬁeld F , then there exist polynomials M ( x ) and N ( x ) over F such that A ( x ) M ( x ) + B ( x ) N ( x ) = G ( x ) . A.5. Prove: If H ( x ) is any other common divisor of A ( x ) and B ( x ) then H ( x ) divides G ( x ). If H ( x ) also has largest degree, then H ( x ) = cG ( x ) for some nonzero constant c ∈ F . Hence we can say that “the” greatest common divisor of A ( x ) and B ( x ) is unique up to nonzero constant multiples. A.6. Euclid’s Lemma for Polynomials. Let P ( x ) be an irreducible polynomial over F (it cannot be factored into two polynomials of positive degree over F ) and suppose that P ( x ) divides a product F ( x ) G ( x ). In this case, prove that P ( x ) must divide either F ( x ) or G ( x ) (or both)....
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 Spring '11
 Armstrong
 Algebra, Intermediate Value Theorem, Prime number, Complex number, Greatest common divisor, Nicolaus Bernoulli

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