# 3124/13997 Modern Algebra - Spring 2010

## Syllabus

Instructor:
| Henning S. Mortveit | Email:
| henning@vt.edu |

Office I:
| 1111 RBXV, Corporate Research Center | Phone I:
| (231-5327) |

Office II:
| 419 McBryde Hall | Phone II:
| - |

Class hours:
| MWF 9:05-9:55AM | Room:
| McBryde 232 |

Prerequisite:
| Math 3034 or Math 3434 | ||

Office hours:
| MW 10:00AM-11:00PM in Office II - Also by appointment in Office I/II |

**Text/Syllabus:**John Durbin,

*Modern Algebra*, 6

^{th}edition, John Wiley and Sons. We will cover selected material from chapters 1-6.

**Course goals:**Learn basic aspects of mathematical structures such as groups, rings and fields. Increase fluency in the reading and writing of mathematical proofs and arguments.

**Exams:**There will be two in-class exams tentatively scheduled for Wednesday February 24 and Wednesday April 7. The two-hour final exam (Section 09M) is on May 8 from 1:05PM to 3:05PM. The final exam will take place in McBryde 232 unless stated otherwise. If you cannot take an exam at the scheduled time, please let me know as soon as possible and

*before*the exam. A make-up exam will be given for reasons that in my judgment are acceptable.

**Homework:**The course has 12 assignments. Generally assignments will be announced each Friday, and will be due in class the following Wednesday. Changes to this will be announced in class. Late homework will only be accepted if handed in the first class following the due date, but only for half the credit.

*Very important:*The assignments are an integral part of this course. The 12 assignments should be considered a minimal effort, and working through additional problems is strongly encouraged.

**Attendance:**Will be taken, and will be kept for Mathematics Department records. Attendance may be used to adjust the final grade.

**Grading:**Is on a curve. However, 90% will be at least an A-, 80% will be at least a B-,70% will be at least a C-, and 60% will be at least a D-. Each assignment is worth 10 points, each in-class exam 40 points, and the final exam 50 points.

**Honor system:**The University Honor System is in effect for assignments and exams (see http://www.honorsystem.vt.edu). Discussion of class topics among students is encouraged, but the solutions to assignments that you hand in must be your own. All exams are closed-book, closed-notes.

**Students with special needs:**Students with disabilities, special needs or special circumstances should meet with the instructor during the first week of classes to discuss accommodations.

**General Notes:**Falling behind in this course is

*dangerous*, so turn in assignments on time, come to class prepared and take advantage of the office hours. (Also: read Professor Bud Brown's hints for success - available at http://www.math.vt.edu/people/brown/hints.html.)

**Supplementary literature:**

- John B. Fraleigh,
*A First Course In Abstract Algebra*, Addison Wesley. (This book is a little more advanced than Durbin's book.) - P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul,
*Basic Abstract Algebra*, Cambridge University Press. (More advanced than Durbin's book. Has many good examples.) - David S. Dummit and Richard M. Foote,
*Abstract Algebra*, John Wiley. (This is the course book for 4124 Abstract Algebra.) - Thomas W. Hungerford,
*Algebra*, Springer Verlag. (A classic reference. A demanding book.)

## Exams

Solution notes and comments will be posted here.### In-class exam 1: Wednesday, February 24

A PDF copy of the exam with answers can be found here. (Password on syllabus.)### In-class exam 2: Wednesday, April 7

A PDF copy of the exam with answers can be found here. (Password on syllabus.)### Final exam: Friday, May 8

## Assignments

Assignments, solution notes and comments will be posted here.### Assignment 1: (Due Wednesday, January 27)

**Problems:**(10 points)

- Section 1: 1, 3-6, 19-21, 28

**Bonus problem:**(1 point)

- Section 1: 29

**Notes:**The comment regarding a typo in Monday's lecture is for homework 2.

**Solution:**Homework 1

### Assignment 2: (Due Wednesday, February 3)

**Problems:**(10 points)

- Section 2: 8, 16, 21-24

**Notes:**In problem 2.21 there is a typo in 6th edition of the course book. Where it says "but beta is not onto" should (of course) say "but alpha is not onto". As always, you have to justify all your answers for full credit.

**Solution:**Homework 2

### Assignment 3: (Due Wednesday, February 10)

**Problems:**(10 points)

- Section 3: 1, 2, 7, 18, 24
- Section 4: 1, 13
- Section 5: 2, 6, 8, 12, 13, 22.
- Problem: Show that m*n = (mn)/2 defines a binary operation on
the set
*G*of positive rational numbers. Classify this binary operation (associative, commutative), find the identity element, and determine the inverses if there are any. You may use any property of the rational numbers without proof, but you should indicate what properties you are using. Is (*G*,*) a group?

**Bonus problems:**(2 points)

- Section 3: 29
- Section 5: 14

**Notes:**

- In Problem 4.13 you may use the result on associativity from Problem 4.12 without proof.
- In problems 5.2, 5.6 and 5.8 you are asked to determine if some set with a given operation is a group. If you determine that it is not a group, then it is sufficient to point out a single reason why this is so - you do not need to check all other conditions.
- Remember to justify your answers.

**Solution:**Homework 3

### Assignment 4: (Due Wednesday, February 17)

**Problems:**(10 points)

- Section 6: 2, 6, 7, 10, 14.

**Bonus problem:**(1 point)

- Let
*G*be a group with identity*e*. Show that if*x***x*=*e*for all*x*in*G*then the group is Abelian.*Hint:*Consider (*a***b*) * (*a***b*).

**Notes:**

- You must justify your answers.

**Solution:**Homework 4

### Assignment 5: (Due Wednesday, March 3rd)

**Problems:**(10 points)

- 3.31, 5.19, 6.13, 7.13 and 7.14

**Bonus problem:**(2 points)

- 7.22

**Notes:**

- On a problem like 6.13 where you may not see the solution right away, it may be helpful to examine some small examples. Once you see how things work it should be easier to tackle the general case. You must justify your answers.

**Solution:**Homework 5

### Assignment 6: (Due Wednesday, March 17th)

**Problems:**(10 points)

- 9.4, 9.13, 9.21 and 9.22
- 10.8, 10.11 and 10.17

**Bonus problem:**(2 points)

- 10.24

**Notes:**

- You must justify your answers.

**Solution:**Homework 6

### Assignment 7: (Due Wednesday, March 24th)

**Problems:**(10 points)

- 11.3, 11.14
- 12.16, 12.19
- 13.13, 13.14, 13.15

**Bonus problem:**(2 points)

- 13.16

**Notes:**

- If you have the 5th edition of Durbin: Problem 11.3 appears as 11.4. Also, you must justify your answers.

**Solution:**Homework 7

### Assignment 8: (Due Wednesday, March 31st)

**Problems:**(10 points)

- 14.4, 14.20, 14.28, 14.31
- 15.2, 15.4, 15.12, 15.19

**Bonus problem:**(2 points)

- 14.34
- 15.26

**Notes:**

- You must justify your answers.

**Solution:**Homework 8

### Assignment 9: (Due Wednesday, April 14th)

**Problems:**(10 points)

- 16.15
- 17.10, 17.12, 17.27

**Notes:**

- You must justify your answers.

**Solution:**Homework 9

### Assignment 10: (Due Wednesday, April 21st)

**Problems:**(10 points)

- 16.6
- 18.5, 18.13
- 19.1, 19.2, 19.3, 19.4, 19.5, 19.13, 19.14

**Bonus problems:**(3 points)

- 16.17, 17.24, 17.26

**Notes:**

- You must justify your answers. For problems 19.1 through 19.5 a single reason is sufficient.

**Solution:**Homework 10

### Assignment 11: (Due Wednesday, April 28th)

**Problems:**(10 points)

- 21.5, 21.18, 21.28, 21.31
- 22.1, 22.2, 22.10

**Bonus problem:**(2 points)

- 21.26

**Notes:**

- You must justify your answers.

**Solution:**Homework 11

### Assignment 12: (Due Monday, May 2nd)

**Problems:**(10 points)

- 23.1, 23.4, 23.12, 23.13

**Bonus problem:**(2 points)

- 23.15

**Notes:**

- You must justify your answers.

**Solution:**Homework 12

Mon Dec 21 11:04:11 EST 2009