Updated: 12/22/07, 1:20pm.
What is new: Answers to HW 09
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Syllabus

• The book will be Serge Lang "Algebra," 4th edition. (Contents, page numbers and layout are identical to the 3rd edition as far as I can tell. Introduction and references are different.) There will be additional material for which detailed notes will be handed out and posted on this webpage. I will assume that students have already seen groups, rings and fields (Math 30ab) but I will review the basic definitions and theorems.
• We will cover the following topics:
  1. Groups (Chapter I): solvable and nilpotent groups.
  2. Category theory (I.11) plus: representable functors and the Yoneda lemma.
  3. Rings and modules (Chap II, III): fundamental theorem for principal rings.
  4. Tensor products and multilinear algebra (Chap XVI)
  5. Fields and Galois theory (Chap IV,V,VI): Things you didn't learn the first time.
• There will be no quizzes or tests. The grade is based on weekly HW and ``class participation.'' You will get a letter grade for each homework.

Instructor

• Kiyoshi Igusa
• Goldsmith 305
• Office hours (subject to change): MWR 10-11 (before class), MWR 12-12:15 (after class), MW 1:15-2 and by appointment.
• Phone 63062.

Lecture Notes

• Part A: Group Theory (50 pp, complete)
  • Part A0: Table of contents so far (page 0) (continuously updating)
  • Part A1: Preliminaries (pp1-3)
  • Part A2: Solvable groups (pp4-7)
  • Part A3: Group actions (pp8-10)
  • Parts A4: p-groups and A5: Nilpotent groups (pp11-14)
  • Part A6: Sylow theorems (pp15-17) (handed out in class)
  • Part A7: Category theory and products (pp18-22)
  • Part A8: Products of many groups (pp23-26) (handed out in class)
  • Part A9: Universal objects and limits (pp27-29)
  • Part A9b: Universal objects and limits: examples (pp30-33)
  • Part A10: Limits as functors (pp34-38)
  • Parts A11, A12: Colimits of groups (pp39-48)
• Part B: Rings and Modules (52 pages, complete)
  • Part B0: Table of contents so far (page 0) (continuously updating)
  • Part B1: Rings and Modules: preliminaries (pp1-3)
  • Part B2: Examples (pp4-8)
  • Commutative rings
    • Part B3: Commutative rings: Basics (pp9-11)
    • Part B4: Localization (pp12-17)
    • Part B5: Principal rings (pp18-19)
  • Modules
    • Part B6: Modules: Introduction (pp19-22)
    • Part B7: Products and coproducts (pp23-29)
    • Part B8: Finite generation and ACC (pp30-34)
    • Part B9: Modules over a PID (pp35-45)
    • Part B10: Jordan canonical form (pp46-49)
• Part C: Tensor products and multilinear algebra (26 pages, complete)
  • Part C0: Table of contents so far (page 0) (continuously updating)
  • Part C1: Bilinear and quadratic forms (pp0-4)
  • Part C1,2,3: Bilinear forms (pp0-11)
  • Part C4: Tensor product (pp12-21)
  • Part C5: Modules over a PID (pp22-25)
• Part D: Galois Theory (33 pages, complete)
  • Part D0: Table of contents so far (page 0) (continuously updating)
  • Part D1,2: Basics (pp0-10)
  • Part D3: Examples (pp11-18)
  • Part D4: Reed-Solomon code (pp19-24)
  • Part D4.7: proof of decoding algorithm (pp25-28)
  • Part D5: Finite fields (pp29-32)

Homework

• Homework 01 with Answers 01.
• Homework 02 with Answers 02.
• Homework 03 (at end of A6 notes: but skip 3.3.) with Answers 03.
• Homework 04 with Answers 04.
• Homework 05 with Answers 05.
• Homework 06 with Answers 06.
• Homework 07 with Answers 07.
• Homework 08 with Answers 08.
• Homework 09 with Answers 09.