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.

18.702 Spring 2018 - a group in a ring in a field.

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.

Course Assistant: Campbell Hewett.

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 11. 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. Fun 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 SaccĂ  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.
  • March 2: Maximal ideals, existence of primes and Zorn's lemma. Reading: [A] 11.8. There are many resources on Zorn's lemma on the web. Also try Atiyah & Macdonald Chapter 1.
  • March 5: Euclidean domains, primes in the polynomial ring in one variable, the statement of the Nullstellensatz. Reading: [A] 11.8
  • March 7: Proof of Hilbert's Nullstellensatz and ideals in the polynomial ring. Reading: [A] 11.8. Fun Reading: An interesting proof the Nullstellensatz by Terry Tao.
  • March 9: Proof of the Nullstellensatz, radical ideals, and affine varieties. Modules. Reading: [A] 14.1-14.6 and class notes. Fun Reading: What is... a syzygy?
  • March 12: Modules. Definitions and examples, free modules, submodules, homomorphisms, and quotients. Reading: [A] 14.1, 14.2. Also read 14.3.
  • March 14: Hom, dual for modules. Reading: Class notes. I will follow Dummit and Foote's Abstract Algebra Section 10.3 but you don't need that book.
  • March 16: Noetherian rings, the Hilbert basis theorem and its proof. Reading: [A] 14.6.
  • March 19: Chain conditions and finite generation. Construction of the tensor product. Reading: [A] 14.6.
  • March 21: Examples of tensor products, working with them. Symmetric products. Reading: Working with tensor products. Fun Reading: What is "canonical"?
  • March 23: Modules over principal ideal domains, classification of finite abelian groups. Reading: [A] 14.7. We will present things in a slightly more general fashion than the book.
  • March 23 to April 2 is break. No assignments or reading will be due during this period.
  • April 2: A review of where we've come with regards to rings and modules.
  • April 4: In class quiz. This will cover the material on rings and modules. Some Practice Questions.
  • April 6: Class will be taught by Davesh Maulik because Dhruv will be on a plane to Texas. Introduction to fields. Reading: [A] 15.1-15.2.
  • April 9: Field extensions, degrees of field extensions, and quadratic extensions. Reading: [A] 15.3.
  • April 11: Random asides, multiplicativity of the degree for field extensions, examples. Reading: [A] 15.3-15.4.
  • April 13: Adjoining elements and finite fields. Reading: 15.6-15.7. Note: We will not cover rule and compass constructions, but this is beautiful mathematics. Read [A] 15.5!
  • April 18: Splitting fields, multiple roots, and the formal derivative. Reading: [A] 15.7.
  • April 20: E. Artin's primitive element theorem and its proof. Reading: [A] 15.8.
  • April 23: A rapid introduction to finite fields -- details will be left out! Reading: [A] 15.8. Fun reading: The field with one element contains two elements!
  • April 25: An overview of Galois theory, the bamboozlement of Galois theory's "main theorem". Reading: Skim [A] Chapter 16. Fun: Try to learn what a covering space is!
  • April 27: Symmetric functions and the definition of a splitting field. Reading: [A] 16.1, 16.2. Fun reading: Cyclotomic fields and Fermat's last theorem.
  • April 30: Criss-crossed towers -- Galois groups and fixed fields. Reading: [A] 16.4, 16.5.
  • May 2: Proof of the main theorem of Galois theory. Reading: [A] 16.6, 16.7.
  • May 4: More on the main theorem of Galois theory. Reading: [A] 16.6, 16.7.
  • May 7: Cubic and quartic equations and their Galois groups. Reading: [A] 16.8, 16.9.
  • May 11: In class quiz. This will cover the material on field extensions and Galois theory only.
  • May 14: A rapid introduction to schemes and the spectrum of a ring.
  • May 16: The p-adic numbers via analysis, Hensel's lemma, Ostrowski's theorem, and the Hasse-Minkowski theorem.


  • Quiz 1 Solutions: A sketch of solutions to the first quiz.
  • Quiz 1 Rewrites: I will offer you the chance to rewrite the quiz for up to 75% of the credit, subject to the following conditions. (1) You must turn in both your original quiz and the rewrite to me by Wednesday March 14. (2) For each problem you rewrite, you must give a short explanation of what went wrong if you attempted a solution. You must then write out a textbook solution to the problems. (3) The level of precision and rigour expected in the rewrites will be substantially higher than on the original timed test, and much higher than the sketched solutions in solutions above. Every sentence must be justified, written clearly, and every step explained fully. The goal is to truly turn the mistakes that were made into a learning experience, and the high standard is maintained to incentivize this. The new grade will replace your original grade on the quiz.
  • Quiz 2 Solutions: A sketch of solutions to the second quiz.
  • Quiz 2 Rewrites: You may rewrite the quiz for up to 75% of the credit. If you got a 6/10 or less on the second quiz I strongly recommend that you take this option. The same conditions hold as before. (1) Your rewrite must be turned in directly to me in class on Friday April 27. (2) Each solution must be accompanied by a short explanation of what went wrong. (3) I expect the solutions to be essentially perfect, see above. The new grade will replace your original grade.

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.
  • Problem set 4: We gain some more familiarity with ideals, explore an example of a non-commutative ring from representation theory, and setup some operations on ideals.
  • Problem set 5: We gain some familiarity with modules, take a look at the localization operation, explore its properties, and the behaviour of primes under localization.
  • Problem set 6: This one explores hom and tensor, introduces a concept called a "valuation ring", and gets a little deeper into the Noetherian property.
  • Problem set 7: This problem set is meant to give us some experience in working with field extensions, and try to observe the range of behaviour that one sees.
  • Problem set 8: We explore fields more, get a sense for finite fields, construct algebraic closures, and begin to inspect Aut(K/F), the basic object of Galois theory.
  • Problem set 9: Shorter than usual. Prove the primitive element theorem for finite fields, give examples of field extensions and Galois groups, and play with symmetric polynomials.
  • Problem set 10: Final problem set! We do some hands on but simple Galois theory computations, and introduce inverse limits and the p-adic integers.