Topics in Algebra
01/22/2007 - 05/05/2007
Tuesday 06:00 PM - 08:30 PM
Science Tower 139
We will begin with a review of such properties of the integers (the whole numbers, positive, negative and zero) as the division algorithm, Euclid's theory of greatest common divisors, unique factorization into primes, etc. We will then see how these results can be extended to polynomials, and examine some of the applications that these extensions make possible such as the determination of the nature of the roots of a polynomial, the derivation of techniques for locating the roots, and the theory of partial fractions.
On a more subtle plane, there is the question of the influence of the choice of the domain over which the coefficients are allowed to vary on the theory of the polynomials in question. Along these lines, we will provide an introduction to Galois Theory and a glimpse of algebraic number theory. Applications here will be made to the questions of solutions of equations by radicals, determining Pythagorean triples, expressing integers as sums of two squares, etc.
A second main topic will be an introduction to transformational geometry, the study of rigid motions in the plane. Application here will be made to congruence and similarity of geometric figures, a classification of the conic sections and a foundation for trigonometry.
These topics can be discussed very fruitfully on many different levels. We will make every effort to choose a level appropriate for those in attendance.
A syllabus for this course is available at:
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
|Level: GLSP||Credits: 3||Enrollment Limit: 18|
Texts to purchase for this course:
NO TEXT REQUIRED
A set of notes will be provided during the first class meeting
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