Math 250A, Algebra

Fall 2021

MWF 10-10:50 AM, MSB 2112

About the course:

The 250 series is meant to equip the graduate student with a thorough background in Algebra, in particular in preparation for the preliminary exam. For 250A, this will include, very roughly, various standard important theorems on groups, fields, and Galois theory.

The assumption is that students taking this course have a solid background in undergraduate algebra. Another assumption is that students will read about the basic notions in the text, and digest these notions, as well as those covered in class, through weekly homework assignments. Being able to read up on topics independently is an important skill every graduate student should work to develop. There is a lot of material to cover, and so class time will be devoted only to main theorems and applications thereof: for example, definitions which you should either have seen in undergraduate algebra or could easily grasp by reading your favorite Algebra book will not be covered during class time. On the other hand, we will cover some material in class that is either not emphasized in most standard books or deferred to exercises.

To make it easier for you to prepare for the lectures, I will post in advance a weekly list of concepts we will cover in class, as well as notions that I will assume you know throughout each lecture on this website.

• NOTE: If you cannot come in person, please know that I will be making avavailable videos of lectures from last Fall (which was taught remotely) on Canvas. I encourage you to please stay home if you are experiencing symptoms or have had an exposure to a known positive COVID case, ideally until you have a negative COVID test 5 days after exposure, or until symptoms resolve.
• Please follow campus regulations and wear a good quality face covering during class. Everyone, including me, will be very thankful to you. I personallyh live with two unvaccinated children and interact daily with high risk elderly parents.

There is no required text for this class. Some books you may consider learning from alongside the lectures are below. If you are to purchase a new book, I would purchase Dummit and Foote.

• Algebra by Artin
• Algebra by Birkhoff and MacLane
• Abstract Algebra by Dummit and Foote
• J.S. Milne's notes on group theory and Galois theory on this website
• Some notes from my course from Fall 2020, beautifully typesetted by graduate student Gregory DePaul.

Grades will be based on weekly homework assignments (40%), one two-hour midterm (25%), and one three-hour final (35%). Dates and times TBA. Most homeworks (unless otherwise noted) will be due on Fridays on Canvas (submitted online) at 11:59PM. Collaboration on the homework is encouraged, but please do cite any sources which helped you to complete the assignment. No late homeworks will be accepted!

• Cyclic Groups by Arthur Ogus
• Simplicity of A_n by George Bergman
• Simplicity of PSL_2(R)
• On the number of Sylow subgroups of a finite group by Marshall Hall.
• G. Bergman's notes on a PID which is not a Euclidean Domain
• Notes on geometric constructions

• 9/22,9/24,9/27: Isomorphism theorems, exact sequences, composition series. This week's suggestions on what to know before class are here.
• 9/29,10/1: Composititon series and Jordan-Holder theorem; solvable groups. This week's suggestions on what to know before class are here.
• 10/4,10/6,10/8: Solvable groups, derived series, automorphism groups, semidirect products. Starting Sylow theorems next week. The next two weeks' suggestions on what to know before class are here.
• 10/11, 10/13, 10/15: Group actions,Sylow theorems. Here are some more suggestions of what to know before class, including some suggestions on field theory, which is coming up.
• 10/18, 10/20, 10/22: Finishing Sylow theorems, beginning field extensions: composite fields, splitting fields.
• 10/25, 10/27, 10/29: More on splitting fields, normal extensions, and algebraic closure.
• 11/1, 11/3, 11/5: Separability and positive characteristic, finite fields. Possibly starting Galois theory. Good news: I will not assume you know anything in particular that we have not alreeady covered throughout the rest of the quarter.
• 11/8: Midterm, 9AM-10:50AM.Some practice problems are here.
• 11/10, 11/12: Dedekind's lemma, Artin's theorem, and several equivalent definitions of a Galois extension.
• 11/15: Galois correspondence theorem.

• Homework 0: send me an email telling me a little about yourself: who you are, what kind of math you like/don't like (be honest, I won't be offended), what your hopes and worries are for this course, and so on.
• Homework 1 due on Friday, 10/1.
• Homework 2 due on Friday, 10/8.
• Homework 3 due on Friday, 10/22. Note extended due date!!
• Homework 4 due on Friday, 10/29.
• Homework 5 due on Friday, 11/12. Notice the later due date due to our midterm on 11/8.
• Homework 6 due on Wenesday, 11/25.
• Homework 7 due on Friday, 12/3.
• Here are some problems to help you study for the final, which is on 12/9, 11AM-2PM.