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# INTRO MDRN ALG MATH 411

UW

GPA 3.76

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This 2 page Study Guide was uploaded by Addison Beer on Wednesday September 9, 2015. The Study Guide belongs to MATH 411 at University of Washington taught by Staff in Fall. Since its upload, it has received 34 views. For similar materials see /class/192109/math-411-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15

Mathematics 41 1 11 December 2002 Final exam preview Instructions As always in this course clarity of exposition is as important as correctness of mathematics 1 N E Recall that a Gaussian integer is a complex number of the form a bi where a and b are integers The Gaussian integers form a ring Zi in which the number 3 has the following special property Given any two Gaussian integers r and s if 3 divides the product rs then 3 divides either r or s or both Using this fact prove the following theorem Given any Gaussian integers r1 r2 divides at least one of the factors ri rquot if3 divides the product r1 in then 3 a Use the Euclidean algorithm to nd an integer solution to the equation 7x 37y l b Explain how to use the solution of part a to nd a solution to the congruence 7x El mod 37 c Use part b to explain why 7 is a unit in the ring Z37 Given a Gaussian integer t that is neither zero nor a unit in Zi a factorization t rs in Zi is nontrivial if neither r nor 3 is a unit a List the units in Zi indicating for each one what its multiplicative inverse is No proof is necessary b Provide a nontrivial factorization of 53 in Zi Explain why your factorization is nontriv ial c Suppose that p is a prime number in Z that has no nontrivial factorizations in Zi Prove that the equation x2 y2 p has no integer solutions Mathematics 41 1 Palmieri 4 Suppose that R is a ring with additive identity 0 An element of r of R is called a zero divisor V39 9 7 if r is nonzero and if there is a nonzero element 3 of R such that rs 0 a Describe a zerodivisor in Z 10 and explain why it is one b Suppose that m is an integer greater than 2 that is not a prime Describe a zerodivisor in Zm and explain why it is one c Suppose that p is a prime number Recall that if p divides a product ab of two integers then it divides at least one of a and b Use this to prove that there are no zerodivisors in Z p One can construct a ringR with six elements R 61 b c d ef with multiplication table as follows a Which element of R is the multiplicative identity Why b Which elements of R are units Why c Is R a eld Why or why not d Can the multiplication table above be the multiplication table of Z 6 with the six ele ments of Z6 being assigned somehow the names a b c d e and f In this problem 1 represents the Euler phifunction and m is a positive integer a De ne b Suppose m pe for some prime number p and positive integer 6 State a formula for in terms of p and e and prove that the formula is correct c State Euler s theorem Make sure that all of the terms occurring are clearly identi ed and any restrictions on them are described d Use Euler s theorem to nd the smallest positive integer c such that 5773 E 6 mod 121 Explain what you are doing in your calculation pointing out in particular how Euler s theorem is used When was Herman Melville born In what city

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