Abstract Algebra -- Math 701

Frank Thorne - Fall 2021

University of South Carolina

Instructor: Frank Thorne, Coliseum 1022D, thorne [at] math [dot] sc [dot] edu.

Classes will be 12:00-12:50, MWF, in [Room TBA]. As the Covid situation dictates, they will also be live-streamed through Microsoft Teams. In-person attendance is generally expected.

Covid-19 Safety:

If you suspect Covid-19, the university requests that you go through official channels: contact the COVID-19 Student Health Services (SHS) nurse line (803-576-8511), complete the COVID-19 Student Report Form and select the option allowing the Student Ombuds to contact your professors. When talking with the SHS nurse, be sure to ask for documentation of the consult as you will need this to document why you missed class. You will also use the COVID-19 Student Report Form if you have tested positive for COVID-19 or if you have been ordered to quarantine because of close contact with a person who was COVID-19 positive. In each of these situations you will be provided appropriate documentation that can be shared through the Student Report Form.


No attendance policy will be enforced, but consistent attendance -- in person, except as dictated by the Covid situation -- is wise.

Course objectives and learning outcomes:

In Math 701 and 702, successful students will:

Office Hours:


The course will use Dummit and Foote's Abstract Algebra, please obtain a copy of the book and follow along.

There are a number of other excellent books on algebra as well. Perhaps the most interesting is that by Lang. In my opinion Lang's book is not very good for the beginner, but is excellent for someone who has seen the material before and wants to review or see a different perspective. In particular, it is great to read when you are studying for quals.

Microsoft Teams:

The course will heavily use Microsoft Teams for announcements, discussion, file sharing, and chat. Lectures will be livestreamed over Teams and recorded, and lecture notes will be uploaded there as well.

Please either check Teams daily, or have it forward new announcements to your email. You can ask me questions directly via Teams. Although private questions are okay, please ask your question publicly if you are comfortable doing so -- maybe others will be interested in the answer as well!

Course Requirements:

The exams will be take-home and pledged, closed book/notes, 4 hours for both the midterm and the final, and similar to what will appear on the qualifying exam. The midterm will be passed out on a date to be announced, and you will have a week to do it. The final exam will be distributed on a date to be announced, and due the last day of finals period.

For the seminar reports you are asked to attend at least four seminars, conference talks, or colloquia and write with a brief report of what you learned, what you found interesting, or what questions you have. The format is up to you; Ravi Vakil's three things is one good choice. You may write privately, but posting publicly to the Teams chat is encouraged.

Seminars may be in-person or online. USC-sponsored seminars include Algebra, Geometry, and Number Theory and the department colloquium. There are also many external online seminars, for example the Number Theory Web Seminar.

You should periodically go to these if a topic catches your interest. Even if you expect the seminar to go over your head, it's worthwhile to go anyway: in part, we learn mathematics the same way babies learn language. Here is Ravi Vakil's advice on how to go to a talk.

Special Accommodations:

If you have a disability, have particular Covid-related concerns, or otherwise need special accommodations, please be in touch with me and I will do my best to accommodate your needs.

Grading Scale:

Homework Assignments:

To be posted, either here or on Microsoft Teams.

Rough schedule of topics:

This is for both 701 and 702, and is very much subject to change.

The course will include a short unit on linear algebra at the very beginning of the term. This will showcase the themes of the course in a setting that will be very familiar, and yet possibly unfamiliar at the same time. (Matrices will be mentioned only briefly if at all, and they will certainly not be row reduced at the blackboard.)

For a more highbrow perspective, I recommend Lang's book. For a still more highbrow perspective, read this or this if you dare.

If there is additional time (wishful thinking?), we will cover additional topics such as homological algebra and group representation theory (see Ch. 17-19 of Dummit-Foote). Another option is to cover the basics of category theory, borrowing from Aluffi's Algebra: Chapter 0. Still another option is to do more noncommutative ring theory. If there are topics you would like to see covered, please be in touch! Other topics may be covered as well, depending on student interest.