Math 653: Graduate Algebra I       Autumn 2025

Instructor: Frank Sottile

Lectures: TuTh 12:45–14:00 Blocker 121

Office: Blocker 601 L in the geometry centre

Office Hours :	Mondays:	14:00–14:50
	Tuesdays:	14:10–15:00
	By appointment

Email: sottile@tamu.edu Text-only with 653 in the subject line.

Book: Algebra by Serge Lang. Download chapters or purchase inexpensive softcover from Springer. (You need to do this on campus.)

Course Description This is a first semester graduate course in abstract algebra, and is intended to be an introduction to the fundamental objects of groups, rings, modules, fields, and vector spaces. I intend to cover all of Chapters I and II, and most of IV from Lang's classical algebra text (at right). We should cover the following topics, time permitting. monoids basic group theory cyclic groups finitely generated abelian groups Sylow theorems categories and functors free groups and inverse limits basic ring theory commutative rings polynomial and group rings localization principal ideal domains and unique factorization domains polynomials in one variable factorization of polynomials Hilbert's theorem partial fractions symmetric polynomials Prerequisites: Undergraduate abstract algebra (Math 415/6) or its equivalents, as determined by the instructor.

Graduate Work:

This is the first of a two-term sequence on Graduate Algebra, which is a foundational course covering material that every mathematician should know. It forms the air one breathes in many research fields, from algebra to number theory, geometry, algebraic geometry, and algebraic combinatorics, and is assumed in many advanced classes in these topics. Basic algebra is fundamental to any educated mathematician. The pace will be fast and advanced for many, for there is a lot to cover. Also, as a graduate class, significantly more is expected of you than in your previous courses.

Reading:

You should read the sections of the book before they are covered in class, work over your notes after lecture (I will not review material covered in previous lectures!), and reread the sections after they are covered. Rewriting your notes is an excellent path towards full understanding (I did this in all of my graduate courses.) Some material that you are responsible for will not be covered in class, but may be found in the book and in the exercises—read those too.

Homework:

There will be weekly written homework assignments to be handed in (on Thursdays, at the start of class) and graded. Problems may be assigned in nearly every class; a summary may be found here. These will be marked by our grader, Daniel Dale, an advanced senior graduate student, and returned on Tuesdays. There may be homework you are to hand into Frank.
As this is a qualifying class, there will be plenty of homework, and you should use it as a way to master the material, which as noted is fundamental to any educated mathematician.

Note that late homework is not accepted.

Zeroth homework: Read this page and the course syllabus, found in Canvas, and send a note to Frank (by email, 653 in the subject line) that you have read and understood the syllabus and course homepage and pledge to abide by the course rules as detailed in those documents. Please also include in that a brief description of who you are, your mathematical background, what you hope to get out of the course, and anything else that you want me to know about you.

Because I understand the pressures that advanced undergraduates and graduates face, I will drop your lowest homework score.

Writing Mathematics:

Developing your ability to write mathematics well is a critical skill for your future studies, and careful writing is linked to clear thinking. Clear, crisp, and correct writeups of problems and proofs are to be expected and should be your goal. For example, after working out the solution to a homework problem, which is at best a very rough draft, you should then neatly write up the solution properly, omitting needless steps and falsehoods (remember, a false claim in a proof invalidates the proof and results in a score of zero), and striving for clarity and brevity.

Group Work:

You are encouraged to work together to find solutions to the homework. However, you must write up the work you turn in yourself. Also, you are absolutely forbidden to consult web sites or problem solution sources (other than the textbook and class notes). Violating this rule has serious consequences, both as it violates the university's academic integrity policy (read that in the official syllabus), and because not doing your own work will likely lead to an inability to do the problems on the course exams and very poor marks. A handy place to collaborate could be Blocker suite 601; it has three boards and a nearby expert or three.

Exams:

The class will have three exams. They will be evening exams during the semester, likely 3 hours each. Sottile will provide dinner. They will be during the weeks of September 24, October 29, and December 3. There is no final exam for this class.

Grading:

Your course grade will be based in equal parts on homework and the midterm exams, That is, 25% homework, and each exam is worth 25%.
I will use the following scale for grades:
A: Above 85%
B: Between 65% and 85%
C: Below 65%
These may be adjusted with lower cut-offs if necessary, as I intend to assign grades that reflect the historical grade distribution.

Device Policy:

I have a zero-tolerance policy towards electronic devices in class. In particular no phones, and no laptops. The exceptions are if you have an accommodation, if we have discussed this before, or if we are using computer algebra software that day. It is OK to use a tablet for notes if you write with a stylus, and pen(cil) on paper is always OK.
Use of an unauthorized device in class will result in your dismissal from class that day.