Math 653: Graduate Algebra       Autumn 2023

Instructor: Frank Sottile

Lectures: MWF 11:30–12:20 Blocker 121

Office: Blocker 601 K in the geometry centre

Office Hours :	Mondays:	14:00–14:50
	Wednesdays:	13:00–14:00
	By appointment

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

Book: Algebra by Hungerford. Download chapters or purchase inexpensive softcover from Springer.

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 most of Chapters I–III and parts of Chapter IV from Hungerford's classical algebra text (at right). We should cover the following topics, time permitting. basic group theory solvable groups finitely generated abelian groups Sylow theorems and basics of the classification of simple groups free groups and inverse limits Rings, integral domains, and fields commutative rings polynomial rings localization principal ideal domains and unique factorization domains power series and power series rings introduction to modules exact sequences free modules and vector spaces Prerequisites: Undergraduate abstract algebra (Math 415/6) or its equivalents.

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, and algebraic geometry, 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.

You are expected to read and understand the first chapter before class starts.

Homework:

There will be weekly written homework assignments to be handed in (on Mondays) and graded. Problems may be assigned in nearly every class; a summary may be found here. These will be marked by our grader, C.J. Bott, an advanced senior graduate student, and returned on Fridays. Sometimes, there will 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 is fundamental to any educated mathematician.

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 striving for clarity and brevity. To facilitate this development, I will assign and collect some problems which I will mark, using a significantly more rigourous scale than perhaps that employed by our grader, which will be discussed in class.

Group Work:

You are encouraged to work together to find solutions to the homeworks. 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 universities academic integrity policy here), and because not doing your own work will likely lead to an inability to do the problems on the course exams.

Exams:

The class will have three exams. There will be two evening exams during the semester, likely 3 hours each. The first will be during the first week of October, and the second will be the week before Thanksgiving. There will also be a final exam Tuesday, 12 December 10:30–12:30.

Grading:

Your course grade will be based in equal parts on homework, the midterm exams, and on the final exam. (One-third homework and final, and one-sixth each of the midterm exams.)