Math 210A: Modern Algebra

 Lectures: Tuesdays and Thursdays 11-12:15 in Rm. 380W.

Office hours: in 380-383M (third floor of the math building). Tuesdays and Thursdays 2:30-3:30. Let me know if these aren't convenient. Pokman and I will have bonus office hours before the midterm and final.

Textbook: Rotman's Advanced Modern Algebra. The bookstore might already be out; if that's the case, please let me know. I am grateful to Stavros Toumpis for pointing out that errata for the book are available at Rotman's homepage.

Grading scheme:

  • 7-8 problem sets (with lowest score dropped) 40%.
  • In-class mid-term 20%.
  • Final exam 40%.

    Course assistant: Pokman Cheung, pokman@stanford.edu. Pokman will be having problem sessions (a.k.a. office hours) in his office, 380-R Mondays 4:30 to 5:30 (note change!) and Wednesday 2-3.

    Problem sets: Problem sets will be due on Tuesdays at 3:30 pm, in Pierre's mailbox (on the first floor of the math building 380, labeled "Albin"). The grader is Pierre Albin, pierre@math.stanford.edu.

    This is the first course in a three-part sequence with the following description from the course guide: "Groups, rings, and fields, Galois theory, ideal theory. Introduction to algebraic geometry and algebraic number theory. Representations of groups and non-commutative algebras, multilinear algebra. Prerequisite: 120 or equivalent."

    We will focus on groups, rings, and fields (including Galois theory), covering roughly to the end of Chapter 5 in Rotman's Advanced Modern Algebra, although not necessarily in order.

    For quals information, click here.

    Bayle Shanks has kindly set up a web site with some of his notes from the course here.

    The course so far: Handouts in dvi, ps, and pdf formats are below. I suspect that the ways I make pdf files are device-dependent (i.e. they may look funny on your machine), so I've tried two different ways (pdfA and pdfB). If you try both and one is better than the other, please let me know. I think pdfB may be better on some machines.

  • The introductory handout (dvi, ps, pdfA, pdfB).
  • Class 1 (Th Sept. 28): Introduction, definitions.
  • Class 2 (Tu Oct. 1): Lagrange's theorem, homomorphisms of groups, conjugacy classes.
  • Class 3 (Th Oct. 3): Normal subgroups, three isomorphism theorems, the correspondence theorem, action of a group on a set.
  • Class 4 (Tu Oct. 8): Actions of a group on a set. Problem set 1 out (dvi, ps, pdfA, pdfB).
  • Class 5 (Th Oct. 10): More group actions; simple groups.
  • Class 6 (Tu Oct. 15): Introduction to commutative rings. Problem set 1 due. Problem set 2 out (dvi, ps, pdfA, pdfB).
  • Class 7 (Th Oct. 17): Properties of rings; PIDs, EDs, UFDs.
  • Class 8 (Tu Oct. 22). PID implies ED. Rings of integers in quadratic number fields. Ring homomorphisms. Problem set 2 due. Problem set 3 out (dvi, ps, pdfA, pdfB).
  • Class 9 (Th Oct. 24). The Chinese Remainder Theorem. Polynomials over a domain, especially a UFD (Gauss' Lemma, R is a UFD iff R[x] is, and more).
  • Class 10 (Tu Oct. 29). Irreducibility criteria (rational root theorem, Eisenstein's criterion). The classification of finite abelian groups. Problem set 3 due.
  • Class 11 (Th Oct. 31). Proof of the Fundamental Theorem of Finite Abelian Groups. Problem set 3.5 out (not to be handed in) (dvi, ps, pdfA, pdfB).
  • Class 12 (Tu Nov. 5). Sylow Theorems and applications. Practice midterm out (dvi, ps, pdfB). Problem set 4 out (dvi, ps, pdfA, pdfB).
  • Class 13 (Th Nov. 7). Midterm (in class, closed book). (dvi, ps, pdfA, pdfB). Topics covered: Chapter 1 (assumed). Chapter 2. Chapter 3 (3.7 linear algebra assumed), except for the parts of 3.8 to do with finite fields. Chapter 5.1. Chapter 6.1 and 6.2. (We've done parts of 6.3 and 6.4, but you're not responsible for them.) Here are the midterm solutions: (dvi, ps, pdfA, pdfB, pdfC).
  • Class 14 (Tu Nov. 12). The Jordan-Holder Theorem. Problem set 4 due.
  • Class 15 (Th Nov. 14). Semi-direct products. Problem set 5 out (dvi, ps, pdfA, pdfB).
  • Class 16 (Tu Nov. 19). Examples of semi-direct products. Galois theory. Problem set 5 due. Problem set 6 out (dvi, ps, pdfA, pdfB, pdfC).
  • Class 17 (Th Nov. 21).
  • Class 18 (Tu Nov. 26). Problem set 6 due. Problem set 7 out (dvi, ps, pdfA, pdfB, pdfC). Here is a proof of the theorem I was in the process of explaining today: (dvi, ps, pdfA, pdfB, pdfC).
  • Class 19 (Tu Dec. 3). Problem set 7 due.
  • Class 20 (Th Dec. 5). Handout on the fundamental theorem of galois theory (dvi, ps, pdfA, pdfB, pdfC). Practice problems for the exam, and Galois theory practice problems (dvi, ps, pdfA, pdfB, pdfC).
  • I recently received a proof of the transcendance of e and pi in the form of a nice series of problems: ps, pdf.
  • Final exam (We Dec. 11, 7-10 pm). There will be bonus office hours on Dec. 10 (me 9-12 am, Pokman 2-5 pm) and Dec. 11 (me 9-12 am). Here is a list of what we've covered in Rotman (dvi, ps, pdfA, pdfB, pdfC). If you think I've omitted something or included something we didn't cover, please let me know as soon as possible, so I can let everyone else know too!
    Back to my home page.
    Ravi Vakil
    Department of Mathematics Rm. 383M
    Stanford University
    Stanford, CA
    Phone: 650-723-7850 (but e-mail is better)
    Fax: 650-725-4066
    vakil@math.stanford.edu