Welcome to the course webpage for 18.702 at MIT, taught in Spring 2018. This course is a secondsemester 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 (2241). Meeting time: MWF 111150 (2190). Office hours: Wednesday 24 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 rewrite it and resubmit it. Your original grade will then be replaced by the new one up to 75%. Quizzes: There will be inclass 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 rewrite 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:
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:
