Homework Assignment 5

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