Math 122: Abtract algebra

Akshay Venkatesh, MWF 11--11:50 in 380D.


here is the take home final. This version was updated on Sunday June 8 to correct the hint for problem 6, as in the email announcement. It is due Wednesday June 11 by 12pm. Please note that this is a hard deadline, because I have a short time to grade it.


Summary
The course gives an introduction to modules over rings, as well as the representation theory of finite groups.

Modules are to rings as vector space as to fields, and they form a model for structures that occur all over mathematics. We'll cover three main topics:


Assessment: Midterm (15%) + final (25%) + homework. (60%). The midterm will be held in-class on Wednesday, April 30. Here is practice midterm. The final will be take-home and due on Wednesday June 11.

Text: Continued from the Math 120, 121 series is Abstract Algebra by Dummit and Foote. We will cover chapters 10, 12, 18 in detail, and 19 as time permits.


Office hours: My office hours will be before class, MWF 10-11. The CA is Niccolo' Ronchetti. His office hours will be Tuesday and Friday 6--7:30 pm.


Syllabus (this is approximate -- we might go a little slower or faster).
  1. Week 1, ending April 4: Definition of modules, examples, homomorphisms, submodules, quotient. The universal property of a quotient. (10.1, 10.2).
  2. Week 2, ending April 11: Free modules, direct product, direct sums and their universal properties. Why universal properties uniquely determine a construction. (10.3).
  3. Week 3, ending April 18: More on universal properties and tensor products. (10.3, 10.4). [Note added April 16: we are running ahead of schedule by about two lectures.]
  4. Week 4, ending April 25: Tensor products: examples, existence (sketch). Modules over a PID: statement of the fundamental theorem. Reformulations of the main theorem. Deduction of the structure theorem for f.g. abelian groups and the Jordan canonical form from this.
  5. Week 5, ending May 2: Proof of the theorem about modules over a PID, via row and column operations on matrices. Finiteness properties of a PID: noetherian rings, any submodule of finitely generated is finitely generated. Change of rings (a bit).
  6. Week 6, ending May 9: Representation theory: equivalent perspectives on a representation as an FG module, a homomorphism from G to GL(V), and a linear action of G on a vector space. Maschke's theorem. Tensor products of representations.
  7. Week 7, ending may 16: duals of representations. Characters of irreducibles are orthonormal (not yet proved). The character table and its basic properties; group algebra is a product of matrix algebras. Character table of S_3 and S_4.
  8. Week 8, ending May 23: Proofs of the basic theorem about characters. Wedderburn's theorem (Brian Conrad).
  9. Week 9, ending May 30: Odds and ends, character table of A_5, S_5.
  10. Week 10: More odds and ends.

Homeworks will be posted here.
  1. First homework, due Friday April 11. 10.1 exercises 4, 8, 15, 18, 20. 10.2 exercises 5, 8, 13. 10.3 exercises 1, 2, 7, 8, 15, 27. Bonus question (don't try unless you have time to spare): Let M, N be as in exercise 10.3.24. Prove that the quotient module M/N has no nontrivial homomorphisms to Z (the integers under addition, as a Z-module).

    Here is a hint for the bonus problem. First of all, identify some special elements in M such that any homomorphism M/N-->Z vanishes on these elements. For instance, any element of M whose entries become divisible by higher and higher powers of 2 has this property. Then show that every element of M can be written as a sum of such "special" elements.

  2. Second homework, due Friday April 25.
  3. Third homework, due Wednesday May 7.
  4. Fourth homework, due Wednesday May 21.

    CORRECTIONS: Book problem "18.2.4" should in fact be "18.3.4"! I am sorry for any suffering thus caused. Book problem 11.3.1 is not correct as stated - find the error. (added Monday 19 night: It's been pointed out to me that this varies from edition to edition and is fixed in the third edition of the book). You can assume that the field is the complex numbers and that representations are on finite dimensional vector spaces unless otherwise specified. Finally, in question 6, P should be a HOMOGENEOUS polynomial of degree d.

  5. fifth homework due Friday June 6. In the final problem, you can assume that n is "large enough" (say n>=6).

Akshay Venkatesh
Department of Mathematics Rm. 383-E
Stanford University
Stanford, CA
email: akshay at stanford math