Instructor: Frank Thorne, LeConte 317O, thorne [at] math [dot] sc [dot] edu.

**In Math 701 and 702, successful students will**:

- Master algebraic topics concerning groups, rings, fields, and other algebraic structures.
- More importantly, develop their sense of
*algebraic thinking*, so that they can more easily absorb additional algebraic material in the future. - Lay the foundation for research in topics including algebraic number theory, algebraic combinatorics,
representation theory, algebraic geometry, commutative algebra, algebraic topology, etc., etc., etc.
- Prepare for the Qualifying Exam in algebra.

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.

- (70%) Written homework assignments will be assigned at most weekly.
- (10%) Take-home midterm (dates TBD, approximately late October, you will have a week window to finish.)
- (20%) Take-home final exam (passed out on the last day of class, due at some point in finals period).
- (10% -- Bonus) Regularly attend the
graduate colloquium.
- (10% -- More Bonus) Ask Alicia to
**give a talk**in the graduate colloquium.

There are other research seminars that may be of interest to you. These include algebraic geometry, number theory, combinatorics, and the department colloquium.

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.

**Exams:** The exam questions will be similar to what will appear on the qualifying exam.
Take-home and pledged, closed book/notes, 4 hours for the midterm, 6 hours for the final.

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

(Shamelessly stolen from Matt Ballard.)

**A**: You have a strong chance of passing the qualifying exam if you prepare reasonably. This also represents a strong foundation for learning further related topics.**B+**: With additional effort, you should be able to pass the qualifying exam if you prepare. This represents a partial foundation for learning further related topics.**B**: You have demonstrated some mastery of the subject material, but should put in a lot of additional effort if you want to pass the qualifying exam or take followup courses in algebra.**< B**: You apparently spent the entire semester playing Pokemon Go.

- Homework 1, due Friday, September 15.
- Homework 2, due Friday, September 22.
- Homework 3, due Friday, September 29.
- Homework 4, due Friday, October 13.
- Homework 5, due Monday, October 30.
- Midterm exam with solutions, due Monday, November 6.
- Homework 6, due Monday, November 20.
- Homework 7, due Friday, December 8.
- Homework 8 (Math 702), due Friday, February 9.
- Homework 9, due Friday, March 2.
- Homework 10, due Monday, March 26.
- Homework 11, due Friday, April 20.

- Lectures 1-8: Linear algebra; introduction to groups.
- Lectures 9-24: Remainder of the group theory. (Some missing pages.)
- Lectures 25-34: Ring theory.
- Lectures 35-38: Introduction to module theory.
- Lectures 39-41: Free modules, Hilbert basis theorem.
- Lectures 42-46: Tensor products and multilinear algebra.
- Lectures 46-50: Modules over PIDs, matrix canonical forms.
- Lectures 51-62: Fields and Galois theory.
- Lectures 63-70: More fields and Galois theory.

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

One innovation I'm trying out is 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.

**Linear Algebra**(2-3 weeks): Dummit-Foote, Ch. 11-12; Axler,*Linear Algebra Done Right*.**Group Theory**(~6 weeks): Dummit-Foote, Ch. 1-5.**Ring Theory**(~4 weeks): Dummit-Foote, Ch. 7-9. For the most part we will deal with*commutative rings*.**Module Theory and Tensor Products**(~3 weeks): Dummit-Foote, Ch. 10, 11.5; Atiyah-Macdonald, Introduction to Commutative Algebra.**Field and Galois Theory**(~6 weeks): Dummit-Foote, Ch. 13-14.- Introduction to
**Commutative Rings and Algebraic Geometry**(~3 weeks): Dummit-Foote, Ch. 15; Atiyah-Macdonald. - Introduction to
**Algebraic Number Theory**(2-3 weeks): Dummit-Foote, Ch. 15.3, 16.2-16.3; Neukirch,*Algebraic Number Theory*.