18.702 Algebra II

Welcome to the course webpage for 18.702 at MIT, taught in Spring 2018. This course is a second-semester undergraduate course in abstract algebra, intended for mathematics majors who have completed 18.701. This page will maintain the schedule for the class, as well as a repository for course materials such as problem sets, the syllabus, notes, and references. 

Instructor: Dhruv Ranganathan (2-241). Meeting time: MWF 11-1150 (2-190). Office hours: Wednesday 2-4 PM. You can try to stop by most mornings between 830 and 10 AM. 

Overview: The first half of this course, 18.701, equips students with the basic language of abstract algebra, including linear algebra and group theory. In this second half, we get to see two of the most fundamental places where this theory comes to life -- representations of finite groups and symmetries of solutions to polynomial equations, also known as Galois theory. These ideas are ubiquitous in modern mathematics. Along the way we'll get through the fundamentals of rings 

Textbook: The main text will be Algebra by Mike Artin, denoted [A]. For the representation theory part, I also recommend a beautiful textbook Representations and Characters of Groups by Gordon James and Martin Liebeck.

Problem sets: Homework will be assigned each Friday and due the following Friday. The problem sets will be posted here. This is the core of the class. If you don not do the homework, you will not get anything out of the course except a few funny sounding mathematical words to share with your family/family.

If a problem set has gone particularly badly: I will offer you the chance to re-write it and resubmit it. Your original grade will then be replaced by the new one up to 75%

Quizzes: There will be in-class quizzes on three occasions during the term: February 28, April 4, and May 14. If a quiz has gone particularly badly: I will offer you the chance to re-write it and resubmit it. In this case, I will also ask you to write brief explanations of what went wrong on each problem. Your original grade will then be replaced by the new one up to 75%

What will the quizzes be like? I will use the quizzes simply to make sure that the basic language of what we've been discussing has come across. In this sense, the quizzes are as much as an assessment of me as they are of the students. They are not meant to trip you up. Do the reading, complete the homework, and come to office hours. All will be well!

Grading: Your grade will be based on your homework scores (70%) and the quiz scores (10% each). 

Reading: The most effective method for doing the reading is probably to skim it before class and then do a careful reading a day or two after class, but this will vary based on your learning style. The class will not assume you've done the reading, but the homework may well. 

Lecture schedule:
  • February 7: Introduction. Group representations, subrepresentations. Reading: TY Lam's article Representations of Finite Groups: A Hundred Years, Part I.
  • February 9: Review. Irreducible representations and representations of abelian groups. Unitary representations and the averaging trick. Reading: [A] Section 10.2 and 10.3.
  • February 12: Maschke's theorem on direct sum decomposition. Reading: [A] Section 10.3.
  • February 14: Characters of representations and their basic properties, The orthogonality relations on characters. Reading: [A] Section 10.4
  • February 16: More on characters with examples. Equivariant maps and Schur's lemma. A proof of the orthogonality relations. Reading: [A] Section 10.7 and 10.8.
  • February 20: (Tuesday is Monday) Modules over the group ring. Induction and restriction. Reading: I will provide notes. See also [A] Section 14.1 and 14.2 for modules.
  • February 21: Class will be taught by Giulia Saccà because Dhruv will be at Rutgers. Introduction to rings. Reading: [A] Section 11.1-11.3.
  • February 23: Rings of functions, ideals, quotients, and ideals in quotients. Reading: [A] 11.1-11.4
  • February 26: Integral domains, principal ideal domains and examples. The Euclidean algorithm. 
  • February 28: In class quiz: this will cover the material on representation theory. Don't stress out about this.
What will the quiz be like?

Each quiz will consist of four questions. There will be a mix of conceptual questions, such as deducing 

The first quiz will cover the material on representation theory of finite groups. It is meant to check, to what extent the ideas have settled in your mind, and how comfortable you are manipulating them. The best way to prepare is to do mathematics -- go look in Artin's book and do some problems. That is not to say that the problems will necessarily come from the book, but the problems will test to what extent you have mastered the material, and doing problems is the best way to do that. I see a lot of value in concrete calculations. For instance, I like questions of the flavour "Decompose the regular representation of S3 into irreducible representations", or "Here is a representation and a linear operator. Use the averaging trick to produce an invariant operator", or "What is the induction of the alternating representation of S2 to S3?". You should also know the statements of the main theorems and their consequences. You need not memorize the proof of the main theorems. There will also be some more conceptual questions. For instance, "Prove that every finite subgroup of GL(n,C) is conjugate to a subgroup consisting of unitary matrices." 

Problem sets: 
  • Problem set 1: The purpose is to get you comfortable with the definitions, compute some examples, and understand the importance of simultaneously diagonalizing operators.
  • Problem set 2: The purpose is to get some concrete calculations done with characters, see how the theory generalizes, and develop an important construction called induction. 
  • Problem set 3: This one gives an example of horrible things that happen for representations over finite fields, and examines some basic properties of rings.