Algebra I

Math 653-600

Fall 1997

Instructor: Peter F. Stiller
Office: Milner Hall 215 or Blocker Bldg. 634B
Office Phone: 5-5727 or 2-2905
E-mail: stiller@math.tamu.edu
Office Hours: Fall Schedule

Course Information

Time: TR 2:20 to 3:35
Place: BLOC 105
Text: Algebra 3rd Edition by Serge Lang, Addison Wesley (1993).
Supplementary Material: See the bibliography below
Prerequisites: The equivalent of Math 415-416. The student should also be familiar with concepts from linear algebra.
Grades: Mid-term Exam (25%), Final Exam (25%), Research Project (25%), Problem Sets (25%). The Mid-term Exam, as well as the Final Exam, will be given in class. Homework problems will be assigned on a regular basis. This will include five special Problem Sets which will be turned in for a grade. The Research Project will involve outside reading and an oral report (below) .
Overview: This course will introduce the students to the basic objects of algebra: groups, rings, modules, fields. We will study a number of important topics including the theory of groups, Galois theory, and avanced concepts in linear algebra and representation theory.

Syllabus

Detailed Description: We will cover a substantial portion of the text. Chapters 1-4 will be covered this semester, along with topics from Part Two.

Topics

Definitions of Semi-Groups, Monoids, and Groups
Basic Group Theory
Solvable Groups
Finitely Generated Abelian Groups
Aspects of the Classification of Finite Groups - Sylow Theorems
Free Groups and Inverse Limits
Definition of Rings, Integral Domains, and Fields
Basic Ring Theory and Ideals
Commutative Rings - Chinese Remainder Theorem
Polynomial Rings
Localization
Principal Ideal Domains and Unique Factorization Domains
More on Polynomials and Polynomial Rings - Irreducibility
Systems of Equations and Resultants
Power Series and Power Series Rings
Introduction to Modules

Bibliography:

Hungerford, Algebra, Springer Verlag.

Week #1 September 1-5

Read Chapter 1, pgs. 1-13. Try the following exercises:

Pages 75-82, Problems 1, 2, 3*, 4

Note: Exercises marked with an asterisk are especially recommended.

Week #2 September 8-12

Continue reading Chapter 1, pgs. 13-20. Try the following exercises:

Pages 75-82, Problems 5, 6, 9*

Note: Exercises marked with an asterisk are especially recommended.

Week #3 September 15-19

Continue reading Chapter 1, pgs. 20-25. Try the following exercises:

Assignment #1 (see handout) is due September 25.

Week #4 September 22-26

Continue reading Chapter 1, pgs. 36-42.

Pages 75-82, Problems 13*, 14

Note: Exercises marked with an asterisk are especially recommended.

Week #5 September 29 - October 3

Continue reading Chapter 1, pgs. 42-46.

Pages 75-82, Problems 43, 44, 45

Note: Exercises marked with an asterisk are especially recommended.

Week #6 October 6 - 10

Continue reading Chapter 1, pgs. 25-33.

Pages 75-82, Problem 15

Note: Exercises marked with an asterisk are especially recommended.

Week #7 October 13 - 17

Continue reading Chapter 1, pgs. 33-36.

Pages 75-82:
Problems on p-groups: 20, 21*, 22, 23
Problems on structure of groups: 24 (done in class), 25*, 26 (done in class), 28,29, 30, 31, 34*, 35
Problems on simple groups: 27* (do all other problems first), 41*.

Note: Exercises marked with an asterisk are especially recommended.