I. Number and title of course

Mathematics 601 - Topics in Modern Algebra

A. The study of modern algebra is an essential part of the training of every mathematician. It involves topics which comprise the foundations for advanced courses in practically all branches of mathematics. In addition, the study of modern algebra has applications in chemistry, physics, biology and engineer ing.

B. The concepts of modern algebra are characterizations of generalizations of the number systems currently studied in elementary and secondary schools. Graduate students preparing to teach these classes should have a firm grasp of these concepts.

A. To help familiarize the student with properties of such abstract systems as monoids, groups, rings and ideals, vector spaces and modules, fields, and algebra.

B. To develop the student's ability to prove theorems and to train him in the techniques of exposition.

A. Preliminary concepts

1. Logic in mathematics

2. Sets and classes

3. Relations and functions

4. The integers and induction

5. Order and coordinality. The Axiom of Choice and Zorn's Lemma.

B. Groups

1. Semigoups, monoids and groups. Examples (Cyclic groups,...).

2. Homomorphisms. Subgroups and cosets.

3. Finitely generated Abelian Groups. The symmetric group. Direct product of groups.

4. Actions and the Sylow theorems.

C. Rings and Ideals. Fields.

1. Rings, subrings and ideals. Examples (Polynomial rings,...).

2. Homomorphisms.

3. Integral domains, division rings and fields. Examples.

4. Ideals. Examples and isomorphism Theorems.

5. Factorization in polynomial rings.

6. Ring and field extensions.

7. Brief outline of Galois theory.

D. Vector spaces and modules (Optional topics).

1. Vector spaces and modules. Examples.

2. Homomorphisms.

3. Groups, rings, and modules of matrices. Examples.

4. Exact sequences.

E. Algebras (Optional topics).

1. Algebras of matrices.

2. Algebra of the symmetric group.

Atiyah, M. and Macdonald, I. G. Introduction to commutative algebra Addison Wesley. 1969.

Bourbaki, Elements of Mathematics, Algebra I Addison Wesley (ISBN 0-201-00639-1), 1973.

Broida and Williamson A comprehensive Introduction to Linear Algebra Addison Wesley, 1989.

Chevalley, C. Fundamental Conceps of Algebra Academic Press, 1956.

Hugerford, T. Algebra Springer. Verlag, 1974.

Fraleigh, J. A First Course in Abstract Algebra Addison Wesley, 1982.

Blyth and Robertson Algebra Through Practice Cambridge University Press, 1985.

Herstein, I. N. Topics in Algebra, Blaisdell Publishing Company.

Kaplanski, I. Fields and Rings The University of Chicago Press, 1972.

Ledermann, W. Introduction to the Theory of Finite Groups Interscience Publishers, 1964.

Rotman, J. The Theory of Groups Allyn and Bacon, Inc., 1973.

Sagan, B. The Symmetric Group Wadsworth and Brooks/Cole, 1991.

Waerden, B. L. van der Modern Algebra, 1950.

Lectures, class d scussion, problem assignments, and written examinations.

The successful completion of the course work usually required for an undergraduate major in mathematics. This is necessary to provide a foundation for the understanding of the course content. An introductory course in modern algebra in recommended.

Three semester hours.

This course proposal was examined in accord with recommended procedures and was approved by the graduate faculty of the Mathematics Department on March 13, 1993.
_____________________________ Department Chairman

X. Catalog description

Math 601. Modern Algebra? Groups, semigroups, and monoids. Homomorphisms. subgroups and cosets. Abelian groups. The symmetric group. Actions and the Sylow theorems. Rings, subrings and ideals. Ring homomorphisms. Integral domains. division rings and fields. Ring and field extensions. Galois theory.