# 461hw5 - largest de-gree of A x and B x over some ﬁeld F...

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Math 461 F Spring 2011 Homework 5 Drew Armstrong Reading. Chapter 6 Problems. A.1. Let f ( x ) = a n x n + ··· + a 1 x + a 0 R [ x ]. If n is even , with a n > 0 and a 0 < 0, prove that f ( x ) has at least two real roots. (Hint: Intermediate value theorem.) A.2. Leibniz (1702) claimed that x 4 + a 4 (for a R ) cannot be factored over R . (In modern language, he claimed that x 4 + a 4 R [ x ] is irreducible .) Prove him wrong. (Hint: What are the fourth roots of - a 4 ?) A.3. Nicolaus Bernoulli (1742) claimed in a letter to Euler that f ( x ) = x 4 - 4 x 3 + 2 x 2 + 4 x + 4 does not factor over R . Euler responded (1743) that f ( x ) has roots 1 ± α/ 2 and 1 ± α/ 2, where α = q 2 7 + 4 + i q 2 7 - 4 . Use this information to prove Bernoulli wrong . A.4. Given a polynomial p ( x ) C [ x ] with complex coeﬃcients, we deﬁne its conjugate polynomial p ( x ) by p ( z ) := p ( z ) for all z C . This has the eﬀect of conjugating the coeﬃcients. Prove that the polyno- mial f ( x ) = p ( x ) p ( x ) has real coeﬃcients. For the following problems you should use Proposition 6.10 in the text, which says: If G ( x ) is a greatest common divisor (common divisor with
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Unformatted text preview: largest de-gree ) 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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