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# INTRO MDRN ALG MATH 412

UW

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This 2 page Class Notes was uploaded by Addison Beer on Wednesday September 9, 2015. The Class Notes belongs to MATH 412 at University of Washington taught by Staff in Fall. Since its upload, it has received 35 views. For similar materials see /class/192111/math-412-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15

Mathematics 412 1 February 2006 Midterm preview Instructions For this exam clarity of exposition is as important as correctness of mathematics The actual exam will be closed book no notes or calculators allowed There will be room on the paper to write your answers Since it is a timed exam you can use abbreviations and shorthand and you don t need to use complete sentences as long as I can easily understand what you re saying 1 N L 4 A friend comes to you and asks if a particular polynomial px of degree 25 in F x is irre ducible The friend explains that she has tried dividing px by every polynomial in F x of degree from 1 to 18 and has found that px is not divisible by any of them She is getting tired of doing all these divisions and wonders if there s an easier way to check whether or not px is irreducible You surprise your friend with the statement that she need not do any more work px is indeed irreducible Prove this that is use the fact that no polynomial of degree between 1 and 18 divides px to prove that px is irreducible Do not simply quote a theorem that makes this problem trivial rather provide an argument from scratch using the given information You may use the fact that the degree of a product of two polynomials is the sum of the degrees of the two polynomials A famous theorem of Gauss s says Every irreducible polynomial in RM has degree either one or two Use this to prove Every polynomial of odd degree in RM has at least one real root For your answers to this question do not simply quote and apply a major theorem Rather give proofs from scratch a Must the polynomial xquot 7 14 be irreducible in QM for every n 2 l Justify your answer b Must the polynomial xquot 7 49 be irreducible in QM for every n 2 l Justify your answer Prove that the polynomial 15x47x3 7 4x2 733 does not factor in Zbc as the product gxhx of two polynomials gx and hx whose degrees are both less than 4 You may use theorems for this problem as long as you explain what you re using Mathematics 412 Palmieri 5 LetK be a eld a State Bezout s Theorem for a pair of polynomials ax and bx in K b Prove the statement below Suppose that ax and bx are relatively prime polynomials in K x and ax divides the product bxcx in K Then ax divides Cx You may use Bezout s theorem in your proof If you do be sure to make clear where and how you are using it c Use the Euclidean algorithm to nd a greatest common divisor of x3 l and x5 l in F3

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