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MATH 323: Rings and Modules The University of British Columbia Winter 2015 Term 2 Miljan Brakočević Homework Assignment 5 This problem set is due on Thursday, February 11, 2016 at the end of class. Please staple your assignment. Late homework will not be accepted under any circumstances. E-mail submission will not be accepted. Do the Dummit & Foote (3rd ed) exercises: Section 7.6: Exercises 1, 2, 4, 5, 6, 7 Do the following additional ones: A1. Find all integers such that they give remainders 2, 6 and 5 when divided by 5, 7, and 11, respectively. A2. Let F be a field. (a) Prove that F [x]/(x2 − x) ∼ = F × F. (b) Is it true that F [x]/(x2 ) ∼ = F × F ? Prove or disprove. A3*. Show that for every positive integer n, there exists n consecutive integers such that none of them is a prime power. (Prime power is a positive integer power of a single prime). 1