Course Identification

The course is suitable for first-year graduate students in mathematics and computer science who have studied at least 2 years of undergraduate algebra courses.

This course is intended particularly for computer-science students who wish to broaden their knowledge in algebra, and for mathematics students who are interested in adding to their knowledge a number of interesting examples and applications in algebra.

Only a basic knowledge of linear algebra will be assumed. More advanced material will be defined and reviewed, so that the course can also be considered an introduction to algebraic structures for those who have not previously studied them. The purpose of the course is to study selected fundamental theorems in algebra by investigating key examples of different algebraic structures that make the theory transparent. We investigate commutative rings, fields, vector spaces and groups. We shall look at some applications, particularly in coding theory, that utilize basic algebraic properties of some structures and help obtain a deeper understanding of these properties. We shall also use examples developed in one area of algebra and apply them to another ? for instance, applications of properties of finite fields in vector spaces and in finite group theory, and applications of field theory to number theory.

1. Commutative rings: Introduction. Ideals, quotient rings.
2. Principal ideal domains, polynomial rings.
3. Unique factorization domains, Euclidean domains.
4. The Chinese Remainder Theorem and its applications to public key codes.
5. Field theory: Introduction. Extensions of fields.
6. Cyclotomic fields ( 1) n Q , Galois groups, examples.
7. Structure theory of finite fields.
8. Detailed examples and properties of finite fields.
9. Vector spaces: Vector spaces over fields of characteristic 2.
10. Error-correcting codes and finite fields of characteristic 2.
11. Group theory: Introduction, subgroups, quotient groups. Sylow's theorems.
12. The discrete logarithm problem and its applications to public key codes.
13. Matrix groups and their structure, conjugacy classes in matrix groups, detailed examples.
14. Finite Simple groups and the classification theorem ? survey.
15. Conjugacy classes in the symmetric group.
16. If time permits: The field of the p-adic numbers

Upon successful completion of the course students will be able to:

1. Comprehend the theory of different algebraic structures through the investigation of detailed examples.
2. Use the familiarity with the examples of algebraic structures studied in the course in applications to other areas of mathematics.
3. Apply theorems of algebraic structures to coding theory, number theory and other areas of Mathematics.
4. Demonstrate proficiency in the understanding of algebraic structures and the fundamental theorems in commutative ring theory, group theory, and the theory of fields and Galois theory.

1. N. Jacobson: Basic Algebra I & II
2. I.M. Isaacs: Algebra: A Graduate Course
3. S. Lang: Algebra