**Math 101a (Algebra I)**
**Kiyoshi Igusa**
**Fall 2007**

Updated: 12/22/07, 1:20pm.

What is new: Answers to HW 09

What is next: Combine files, write introduction, clean up website.

**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:
- Groups (Chapter I): solvable and nilpotent groups.
- Category theory (I.11) plus: representable functors and the Yoneda lemma.
- Rings and modules (Chap II, III): fundamental theorem for principal rings.
- Tensor products and multilinear algebra (Chap XVI)
- 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
- Modules

- Part C: Tensor products and multilinear algebra (26 pages, complete)
- Part D: Galois Theory (33 pages, complete)

**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.