Course Guidelines

Abstract Algebra II

University of Puget Sound

Math 491

Spring 2020

Dr. Beezer


We will be using Abstract Algebra: Theory and Applications, by Thomas W. Judson as our textbook. We will cover material from Chapters 16–23, as described on the attached calendar. This is an open source textbook, which in part means you are free to make unlimited copies. The book's website is The “2019 Annual Edition” will be the version I will follow for this course. Do not purchase an older edition, as there have been substantial changes in how theorems and examples are numbered, and it will be difficult to follow along with an older copy.

We will be using an “enhanced” online version this semester, more about that in class.

The book's website has links to help you with the purchase of a physical copy of the book, should you desire one. You may also download a PDF that is nearly identical to the hard copy, or another PDF which contains the extra material about Sage. Additionally, the online version has all the same content and the Sage examples are executable and editable, via the Sage Cell server, so is a far superior way to use the book.

As you begin working with Sage, you could find Gregory Bard's Sage for Undergraduates very useful. It is freely and legally available for download as a full-color PDF. (See links in electronic versions of this syllabus, or on the course page).

Course Web Page

Off of you can find the link to the course web page.

Office Hours

My office is in Thompson 303. Making appointments or simple, non-mathematical questions can be handled via email — my address is I rarely do not receive your email, and I read all of my email all of the time, usually very shortly after receiving it. Urgency of replying varies by the hour, day and nature of the message. Office Hours are 9:00–9:50 on Monday and Friday, 8:00–8:20, 9:30–9:50 on Tuesday and Thursday. Office Hours are first-come, first-served, so I do not make appointments for these times, nor do you need to ask me if I will be present at these times. You may assume I will be there, unless I have announced otherwise in class or by email. You may make an appointment for other times, or just drop by my office to see if I am in. Office Hours are your opportunity to receive extra help or clarification on material from class, or to discuss any other aspect of the course.

Class Preparation

Reading questions will help you prepare for the lectures on each chapter. They are available in our enhanced online version of the textbook.

  1. These are due to be completed by 6:00 AM the morning of the day when we begin discussing a new chapter, as indicated on the schedule and/or adjusted/announced in class.
  2. Under no circumstances will they be accepted late.
  3. You can expect a reply that morning, or within 30 hours at the latest. After that time, an email inquiry is appropriate.

Abstract algebra has become increasingly important for its application to digital technologies. For example, we will discuss more linear algebra (Chapter 20), a key subject for data science and machine learning. Finite fields (Chapter 22) are indispensable for cryptography, the basis for online security. Conversely, digital technologies are an ideal assistant for studying the subject. So computation will be a feature of the course.

For this reason, we will make extensive use of Sage. Since Sage is open source software, it is available freely in many places. You will need to purchase an account at CoCalc where we will have access to a powerful servers via your web browser and we can efficiently manage homework assignments. (Details on accounts will be provided in class, cost is $14 for the semester.) The assumption is that you have a paid membership on CoCalc for doing these assignments, so availability, version incompatibility, or convenience of other sites is not an excuse for not being able to complete the Sage assignments on-time.

For each chapter there will be assigned exercises to work in Sage. These will be due roughly on the discussion day following the lectures for each chapter, as a Jupyter notebook. We will discuss the exact procedure in class. Exact due dates will be announced in class. Under no circumstances will these assignments be accepted late.


Exercises from the text will be suggested for each chapter. Of course, you are not limited to working just these assigned problems and you can find many more in textbooks in the library (ask me for suggestions). We have twelve class days reserved for discussions when we can talk about these problems. It is your responsibility to be certain that you are learning from the homework exercises. The best ways to do this are to work the problems diligently, start studying them early, and participate in the classroom discussion. If at this point you are still unsure about a problem, then a visit to my office is in order, since you are obviously not prepared for the examination questions. Making a consistent effort outside of the classroom is the easiest way (only way?) to do well in this course.

Mathematics not only demands straight thinking, it grants the student the satisfaction of knowing when he [or she] is thinking straight.

―D. Jackson

Mathematics is not a spectator sport.


I hear, I forget. I see, I remember. I do, I understand.

―Chinese Proverb

An education is not received. It is achieved.


Each student will research a new topic related to the entire course (i.e. both 490 and 491) and use this as the subject of a paper and an in-class presentation. Details are distributed separately and will be discussed in class as the dates approach.


There will be four 50-minute timed examinations. Planned dates are all listed on the tentative schedule. The comprehensive final examination will be given at 8 AM on Friday, May 15. The final exam cannot be given at any other time, so be certain that you do not make any travel plans that conflict, and also be aware that I will allow you to work longer on the final exam than just the two-hour scheduled block of time.


Grades will be based on the following breakdown:

Attendance and improvement will be considered for borderline grades, while excessive attendance and late-arrival problems will result in grade penalties. Scores will be posted anonymously on the web at a link off the course page.

Academic Policy Reminders

Here are three reminders about important academic policies which are described thoroughly in the “Academic Policies” section of the University Bulletin. The online version is off of

or a printed copy may be requested from the Registrar's Office (basement of Jones Hall).

Registration for Courses of Instruction, Non-Attendance

“Regular class attendance is expected of all students. Absence from class for any reason does not excuse the student from completing all course assignments and requirements.”

Grade Information and Policy, Withdrawal Grades

Withdrawal grades are often misunderstood. A Withdrawal grade (W) can only be given prior to the university deadline listed on our course schedule, and after that time (barring unusual circumstances), the appropriate grade is a Withdrawal Failing (WF), even if your work has been of passing quality. See the attached schedule for the last day to drop with an automatic ‘W’.

Academic Integrity

All of your graded work is expected to be entirely your own work, this includes Reading Questions and Writing Exercises. Anything to the contrary is a violation of the university's comprehensive policy on Academic Integrity (cheating and plagiarism). Discovered incidents will be handled strictly, in accordance with this policy. Penalties can include failing the course and range up to being expelled from the university.


At this point in your college career, you should be well on your way to being an independent scholar, who appreciates the beauty of mathematics and understands the effort needed to master new and difficult ideas. Consistent with that, I will be giving you a fair degree of freedom to learn this material in a manner that suits you. Of course, with freedom comes responsibility.

Read the book before the lectures, work the exercises early and diligently, tidy up your class notes each evening, and ask questions. Arriving late to class, or having conversations with others during class, not only disrupts your peers, but tells me you are not serious about your education.

“Modern” algebra is the basis of one of the two main branches of mathematics (analysis being the other). So every mathematician should have a basic understanding of its principal concepts. The investment of your time and energy applied mastering its basic concepts will be amply repaid by a full understanding of its deeper ideas.


Daily attendance is required, expected, and overall a pretty good idea. Class will begin on-time, so be here, settled-in and ready to go. In other words, walking in the door at the exact time class is to begin is not considered arriving on-time. Repeated tardieness and absences will result in grade penalties, in accordance with university policies. Do not leave class during the lecture unless your continued presence would be a greater interuption — fill your water bottles, use the toilet, and so on, in advance. I do not care how much food or drink you bring to class, so long as it does not distract others or make me hungry. Please do not offer me sweets. Please keep phones in your pocket or bag, unless you are using them to read course material. In short, we are here to learn and discuss mathematics together. It is your responsibility to not distract your peers who are serious about their education or distract me as I endeavor to make the best use of the class time for you and your colleagues.

University Notices

These are multiple notices the university administration requests we duplicate for you.

Student Accessibility and Accommodation

If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Peggy Perno, Director of Student Accessibility and Accommodation, 105 Howarth, 253.879.3399. She will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential.

I request that you give me at least two full working days to respond to any requests from this office.

Classroom Emergency Response Guidance

Please review university emergency preparedness and response procedures and a training video posted at

There is a link on the university home page. Familiarize yourself with hall exit doors and the designated gathering area for your class and laboratory buildings.

If building evacuation becomes necessary (e.g., earthquake), meet your instructor at the designated gathering area so she/he can account for your presence. Then wait for further instructions. Do not return to the building or classroom until advised by a university emergency response representative.

If confronted by an act of violence, be prepared to make quick decisions to protect your safety. Flee the area by running away from the source of danger if you can safely do so. If this is not possible, shelter in place by securing classroom or lab doors and windows, closing blinds, and turning off room lights. Lie on the floor out of sight and away from windows and doors. Place cell phones or pagers on vibrate so that you can receive messages quietly. Wait for further instructions.

Copyright and Fair Use

Course materials are subject to the copyright law of the United States (Title 17 U.S. Code). They are for educational purposes only and limited to students enrolled in the course. Further reproduction or distribution is prohibited. [RAB: Except where publication includes an open license.]

Student Bereavement Policy

The University of Puget Sound recognizes that a time of bereavement can be difficult for a student. Therefore, the university provides a Student Bereavement Policy for students facing the loss of a family member.

Students are normally eligible for, and faculty members are expected to grant, three consecutive weekdays of excused absences, without penalty, for the death of a family member, including parent, grandparent, sibling, or persons living in the same household. Should the student feel that additional days are necessary, the student must request additional bereavement leave from the Dean of Students or the Dean's designee. In the event of the death of another family member or friend not explicitly included within this policy, a bereaved student may petition for grief absence through the Dean of Students office for approval.

Learning Outcomes

The University Curriculum Committee and accrediting agencies expect to see a list of learning outcomes.

Please review these at the end of the semester when they will be even easier to understand.

Tentative Daily Schedule
Monday Tuesday Thursday Friday
Jan 20
Jan 21
Chapter 16
Jan 23
Jan 24
Jan 27
Jan 28
Jan 30
Jan 31
Chapter 17
Feb 3
Problem Session
Feb 4
Feb 6
Feb 7
Travel Day
Feb 10
Feb 11
Feb 13
Problem Session
Feb 14
Exam 1
Chapters 16, 17
Feb 17
Chapter 18
Feb 18
Feb 20
Feb 21
Feb 24
Feb 25
Chapter 19
Feb 27
Problem Session
Feb 28
Mar 2
Mar 3
Mar 4
Mar 6
Problem Session
Mar 9
Exam 2
Chapters 18, 19
Mar 10
Chapter 20
Mar 12
Mar 13
Tentative Daily Schedule
Monday Tuesday Thursday Friday
Mar 23
Mar 24
Mar 26
Problem Session
Mar 27
Chapter 21
Mar 30
Mar 31
Apr 2
Project Proposal,
due Sunday 4/3
Apr 3
Last Day to
Drop with W
Apr 6
Problem Session
Apr 7
Exam 3
Chapters 20, 21
Apr 9
Chapter 22
Apr 10
Apr 13
Apr 14
Apr 16
Apr 17
Chapter 23
Apr 20
Problem Session
Apr 21
Apr 23
Apr 24
Project Paper,
due Sunday 4/26
Apr 27
Apr 28
Apr 30
Problem Session
May 1
Exam 4
Chapters 22, 23
May 4
May 5
May 7
Reading Period
May 8
Reading Period
Final Examination: Friday, May 15, 8 AM
Suggested Exercises
Chapter Computational Theoretical
16 1, 3, 5, 6, 7, 8, 9, 10, 12 2, 16, 19, 20, 24, 26, 27, 28, 33, 36, 38
17 3bc, 4ab, 5ab, 7, 8, 10, Additional: 2-8 13, 14, 17, 18, 19, 23, 24, 25
18 1, 10, 15 5, 7, 9, 11, 12, 13, 14, 17, 19
19 1, 2, 3, 5, 11 12, 13, 15, 16, 18, 21, 22, 23
20 3, 4, 9 10, 13, 16, 18 (maybe more to come)
21 1, 2, 3bcd, 4, 6, 8, 9 11, 16, 19, 20, 21
22 1bc, 3, 4, 7, 8 14, 15, 17, 18, 21
23 1, 2, 3, 4, 5, 11 6, 7, 9, 12, 13, 14, 20