SELECTED TOPICS IN MODERN ALGEBRA I

MA 622

1.  Course Description

 

This course extends the concepts that are taught in an undergraduate introduction to abstract algebra.  Topics include: finite and infinite groups, rings, ideals, and integral domains and fields.

 

2.  Goals of the Course

 

1. To expand the undergraduate abstract algebra concepts

2. To introduce additional topics such as rings and fields

3. To prepare students for additional mathematics courses, possibly at the doctoral level

 

3.  Instructional Procedures

 

1. Lecture/discussion

2. Small group study

3. Use of computer software

 

4.  Course  Content

 

I. Review of Mappings and Operations

a. Mappings

b. Composition, invertible mappings

c. Operations

d. Composition as an operation

 

II. Review of Groups

a.  Definition and examples

b. Permutations

c. Subgroups

d. Groups and symmetry

 

III. Equivalence, Congruence, Divisibility'

a.  Equivalence relations

b. Congruence, the division algorithm

c.  Integer modulo n

 

IV. Groups

a.  Elementary properties

b. Generators

c. Cyclic Groups

d. Direct Products

e  Cosets

f.  Normal Subgroups

g.  LaGrange's Theorem

V. Homomorphism and Factor Groups

a. Homomorphism

b. Isomorphism and Cayley's Theorem

c. Factor Groups

d. Factor-Group Computations and Simple Groups

e. Groups in Geometry

 

VI. Introduction to Rings

a. Definition and Examples

b. Integral domains, subrings

c. Fields

d. Isomorphism

 

VII. The Familiar Number Systems

a. Ordered Integral Domains

b. The Integers

c. Field of Quotients

d. Field of Rational Numbers

e. Ordered Fields

f.  Field of Real Numbers

g. Complex Roots of Unity

 

5.       Evaluation Measures for Determining Students’ Grades

      1.     Tests and Quizzes                         50%

            2.     Homework assignments and projects        15%

   3.     Final exam                                                35%

           

            Note: Individual instructors may weigh evaluation measures differently.

 

6.       Bibliography

A.      Required Text

Fraleigh, John B. A First Course in Abstract Algebra, 5th ed., Reading, MA, Addison-Wesley, 1994.

 

Note: In mathematics courses it is usually preferable to have a designated textbook which helps to focus the discussion and standardize the language and symbolism.

 

B.      Additional Required Readings

               None

 

C.      Supporting Bibliography

Artin, M. Algebra, Englewood Cliffs, NJ,  Prentice-Hall, 1991.

 

Gallian, J.A. Contemporary Abstract Algebra 2d ed., Lexington, MA, D.C. Heath, 1990.

 

 Hillman, A.P. and Alexanderson, G.L. Abstract Algebra: A First Undergraduate Course, 5th ed., Boston MA, PWS Publishing, 1994.

 

McCoy, N.H. Rings and Ideals, Carus Monograph No. 8, The Mathematical Association of America, 1948.

 

McCoy, N.H. and Janusz, GJ. Introduction to Modern Algebra, 5th ed., Dubuque IA, Wm. C. Brown Publishers., 1992.

 

D.      Relevant Periodical Sources

               None

 

E.       Relevant Software

               Exploring Small Groups