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Unformatted text preview: 56 1. ALGEBRAIC THEMES 1.8.15. (a) Suppose that p.x/ D a n x n C C a 1 x C a 2 ZOEx . Suppose that r=s 2 Q is a root of p , where r and s are relatively prime integers. Show that s divides a n and r divides a . Hint: Start with the equation p.r=s/ D , multiply by s n , and note, for example, that all the terms except a n r n are divisible by s . (b) Conclude that any rational root of a monic polynomial in Z OEx is an integer. (c) Conclude that x 2 2 has no rational root, and therefore p 2 is irrational. 1.8.16. (a) Show that a quadratic or cubic polynomial f.x/ in KOEx is irre ducible if, and only if, f.x/ has no root in K . (b) A monic quadratic or cubic polynomial f.x/ 2 Z OEx is irre ducible if, and only if, it has no integer root. (c) x 3 3x C 1 is irreducible in Q OEx . 1.9. Counting Counting is a fundamental and pervasive technique in algebra. In this sec tion we will discuss two basic counting tools, the binomial coefficients and the method of inclusionexclusion. These tools will be used to estaband the method of inclusionexclusion....
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 Fall '08
 EVERAGE
 Algebra, Integers

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