STATE UNIVERSITY COLLEGE AT BUFFALO

Department of Mathematics

Course Proposal

(Updating Existing Course)

I. Number and title of course

Mathematics 651 Theory of Numbers
II. Reasons for addition to the present curricula
A. The theory of numbers has been a fundamental and fascinating subject for mathematicians and teachers of mathematics for many decades. Number theory is a basic tool in many branches of mathematics.

B. In recent years extensions and generalizations of classical number theory have developed in such areas as modern B algebra and set theory. Algebraic number theory is a fruitful field of research.

III. Major objectives of the course
A. To help the student develop the techniques of solving problems in number theory by doing copious classical problems and newer results.

B. To examine some of the more recent contributions to number theory and some of its generalizations.

IV. Topical Outline
A. Divisibility
1. Definitions
2. Prime numbers
B. Congruencies
1. Solution of congruencies
2. The Euler function
3. Power residues
4. Number theory from an algebraic view point.
C. Quadratic Reciprocity
1. Quadratic residues
2. Legendre symbol
3. Quadratic reciprocity
4. Jacobi symbol
D. Functions of Number Theory
1. Numerical functions
2. Moebuis inversion formula
3. Recurrence functions
E. Algebraic Numbers
1. Algebraic number fields
2. Algebraic integers
3. Quadratic fields
4. Unique factorization
V. Bibliography, texts, and readings
Davenport, H. The Higher Arithmetic. London, Hutchinsons University Library, 1952.

Dickson, L.E. Introduction to the Theory of Numbers. Chicago, U. of Chicago Press, 1929.

Hardy, G. H and E. M. Outright. An Introduction to the Theory of numbers. 3rd. ed. Oxford, Clarendin Press, 1954.

Jones, B. W. The Arithmetic Theory of Quadratic Forms. Carus Monograph 10, New York, John Wiley and Sons, 1950.

Le Veque, W. J. Topics in Number Theory. Vols. I and II, Reading Mass., Addison-Wesley, 1956.

Nagell, T. Introduction to Number Theory. New York, John Wiley and Sons, 1956

Niven, Ivan. Irrational Numbers. Carus Monograph 11, New York, John Wiley and Sons, 1956.

Ore, O. Number Theory and its History. New York, McGraw-Hill, 1949.

Pollard, H. The Theory of Algebraic Numbers. Carus Monograph 9, New York, Dover, 1954.

VI Presentation and evaluation
Lectures, class discussions, assigned problems, papers.
VII. Prerequisites
Graduate standing
VIII. Credit
Three semester hours
IX. Statement of approval
This course updating proposal was examined in accord with recommended procedures and has been approved by the Mathematics Department

____________________________________ (Chairman)

X. Catalog Description
MATH. 651 - THEORY OF NUMBERS - Primes, Congruencies, Euler function, power residues, quadratic reciprocity, numerical functions, Moebius inversion, algebraic integers, quadratic fields, unique factorization.

Prerequisites: Graduate Standing

Credit: Three semester hours

XI. Qualifications
Earned Ph.D. in Mathematics