Math 654: Graduate Algebra II       Winter 2021

Instructor: Frank Sottile

Lectures: MWF 9:10–10:10. Purely On-line
Zoom ID for lectures and office hours: 958 3410 6323. Contact Frank for the password.

Office Hours :	20 minutes after every class meeting
	By appointment

Office: My Kitchen

Email: sottile@math.tamu.edu Text-only with 654 in the subject line.

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

Course Description This is the second semester of a 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 will cover parts (Sections 4 and 5) of Chapter III, and then Chapters IV–VI from Hungerford's classical algebra text (at right). We should cover the following topics. I also expect to cover some topics in Chapters VII–X, time permitting. polynomial rings localization power series and power series rings introduction to modules exact sequences modules over a principal ideal domain free, projective, and injective modules hom and duality tensor, symetric, and exterior products field extensions algebraic, separable, and normal extensions simle extensions, splitting fields, and cyclotomic fields fundamental theorem of Galois theory finite fields solvability by radicals transcendence bases Prerequisites: Graduate Algebra I or its equivalent.

Graduate Work:

This is the second 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. 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 to be graded. (Hand in a .pdf created from a LaTeX source code; it is past time for you to learn LaTeX.) We will be using Gradescope, (join it with Entry Code 86E3WZ). These will be assigned regularly, and may be found here. These will be marked by our grader, CJ Bott, who is an advanced graduate student. All problems have equal value. Sometimes, I will assign a problem to grade myself; this helps me to understand how you are doing.
Homework will be typically due Mondays, and always at 9 AM (before class starts).

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. For example, you should first work out the solution to each homework problem, then revise it, then typeset it in Latex, and revise that again, all before submitting it on Gradescope. Please write up the solution properly, omitting needless steps and falsehoods (remember, a false claim in a proof invalidates the proof), and strive for clarity and brevity. To facilitate this development, I will regularly assign and collect problems which I will mark.

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 university's Honor System Rules, 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 and a final.
The first exam is Monday, 1 March in class. It will cover Sections 3.4, and 4.1—4.4.
The second will be Wednesday, 31 March in class. It will cover Sections 4.5, 4.6, 5.1, and 5.2.
The third will be Wednesday 28 April in class. It will cover material since the previous exam. The final exam is scheduled for 4 May from 8 – 10:30. It is cumulative.

Grading:

Your grade depend upon homework and tests. Every homework problem has nearly same weight, and every test problem has 4 times the weight of a homework problem.