Spring 2006

MTHS 662
Number Theory

Reid,James D.

01/23/2006 - 05/06/2006
Tuesday 05:30 PM - 08:00 PM

Science Tower 139

People have been fascinated by numbers and their properties for thousands of years. Following earlier work of the Greeks, notably Eudoxus and Euclid, the work of Fermat (1601--1665), Euler (1707--1783) and Gauss (1777--1855) developed the theory into a more modern form. At the hands of many others, among them the most talented mathematicians of their times, number theory--in its results and in its connections with other areas--has now become one of the most important and profound branches of mathematics. We will survey the basic ideas of the theory, including the Euclidean Algorithm and its applications; primes and the Fundamental Theorem of Arithmetic; congruences and the theorems of Fermat and Euler; and the quadratic reciprocity law as formulated and proved by Gauss. These ideas and results will be applied to the study of such topics as Wilson's Theorem, Mersenne primes, the Chinese Remainder Theorem, primality tests, and cryptography.

The amount of material covered will depend to a large extent on the background and interests of those in the class. However, beyond the basics we hope to include an introduction to algebraic number theory in the guise of the Gaussian Integers. Results here can be applied to the study of Pythagorean Triples, sums of two squares, and other topics. Throughout the term, background will be developed as necessary in order to be able to treat certain of these topics in some depth.

Mathematics is best learned through active engagement. As a guide for this, problem sets will be assigned weekly. These, together with a short paper will form the basis for a grade in the course. Requirements include one paper and weekly problem sets as assigned.

A syllabus for this course is available at:
Course Syllabus

James Reid (B.S., M.A. Fordham University; Ph.D. University of Washington) is professor of mathematics, emeritus. He has published extensively in the field of Abelian Groups, including recent papers "Quotient divisible groups, omega-groups, and an example of Fuchs. Abelian groups, rings, modules, and homological algebra", 265-273, Lect. Notes Pure Appl. Math., 249, Chapman & Hall/CRC, Boca Raton, FL, 2006; "Endomorphism Rings of Free Modules," Rocky Mtn. J. Math., 2002; and "Some Matrix Rings Associated with ACD Groups," Proc. International Conference on Abelian Groups and Modules, Dublin, 1998.


Consent of Instructor Required: No

Format: Seminar

Level: GLSP Credits: 3 Enrollment Limit: 18

Texts to purchase for this course:
David Burton, ELEMENTARY NUMBER THEORY (McGraw-Hill Publishers) 6th Edition, Hardcover


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