I. Number and title of course

Mathematics 701 - Modern Algebra I

A. In recent years the study of modern algebra has become an essential part of the training of every mathematician. It involves topics which comprise the foundations for advanced courses in analysis and topology. In addition, the study of modern algebra has applications in chemistry, physics, and engineering.

B. The concepts of modern algebra are characterizations or 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 groups, rings, ideals, and fields.

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

A. Preliminary concepts

1. Relations

2. Functions and compositions

3. Isomorphism

B. Groups

1. Cyclic groups

2. Transformation groups

3. Subgroups

4. Normal subgroups

5. Factor groups

C. Group theorems

1. Groups with operators

2. Isomorphism theorems

3. Composition series

4. Direct products of groups

5. Sylow theorems

D. Rings

1. Residue class rings

2. Homomorphism of rings

3. Extension of rings

4. Polynomial rings

Artin, E., C. J. Nestitt, and R. N. Thrall. Rinds with Minimum Condition. HA n Arbor: The University of Michigan Press, 1944.

Bourbaki, N. Elements de Mathematique, Livre II, Algebre. Paris: Hermann, 1942-1959.

Chevalley, C. Fundamental Concerts of Algebra. New York: Academic Press, 1956.

Fuchs, L. Abelian Grouns. New York: Pergamon Press, 1960.

Hall, M., Jr. The Theory of Groups. flew York: The Macmillan Company, 1959.

Jacobson, N. The Theory of Rings. New York: American Mathematical Society, 1943.

Kaplanski, I. Infinite Abelian Groups. Ann Arbor: The University of Michigan Press, 1954.

Kurosh, A. The Theory of GrouDs. New York: Chelsea Publishing Company, 1960.

Ledermann, W. Introduction to the Theorv of Firite Groups. New York. Interscience Publishers, 1961.

Miller, Kenneth S. Elements of Modern Abstract Algebra. New York: Harper and Bros., 1958.

Northcctt, D.G. Ideal Theory. Cambridge: Cambridge University Press, 1953.

Schilling, O. The Theory of Valuations. New York: American Mathematical Society, 1950.

Steinitz, E. Algebraische Theorie der Coercer. New York: Chelsea Publishing Company, 1950.

Zariski, O., and P. Samuel. Commutative Algebra, Vol. I and II. Princeton: D. Van Nostrand Co., Inc , 1960.

Lectures, class discussions, problem assignments, and written examinations.

The successful completion of the course work usually required for an undergraduate major in mathematics. (At least 24 semester hours of undergraduate mathematics) This is necessary to provide a foundation for the understanding of the course content. An introductory course in modern algebra is recommended.

Three semester hours

This course proposal was examined in accord with recommend Procedures and was approved by the graduate faculty of the Mathematics Department on June 2, 1966.

________________________________Department Chairman

MATH. 701. MODERN ALGEBRA I - Cyclic groups; transformation groups; factor groups; groups with operators; isomorphism theorems; composition series; direct products of groups; Sylow theorems; residue class rings; operations on ideals; extensions of rings.

Faculty assigned to teach this course must have a concentration of graduate study in the area of Modern (Higher) Algebra.

NAME PREPARATION EXPERIENCE

Montgomery, Mabel Ph.D., SUNY at Buffalo 24 years

Smith, Sigmund Ed.D., Penn State University 10 years

Stern, Samuel Ph.D., SUNTY at Buffalo 9 years