Summer 2004

### MTHS 658 The Algebra and Geometry of Complex Numbers

Mulvey,Irene

06/28/2004 - 08/11/2004
Monday & Wednesday 09:30 AM - 12:00 PM

Science Tower 137

Not every polynomial has a root in the real numbers; for example, the equation (x^2+1=0) has no solution in the real numbers. But, every polynomial has a root in the complex numbers. This indicates a fundamental algebraic difference between the set of real numbers and the set of complex numbers. The result that every polynomial has a root in the complex numbers is known as the Fundamental Theorem of Algebra and it was proved by Karl Friedrich Gauss in his doctoral thesis in 1799. (Now, there are reportedly over 100 proofs of this theorem and Gauss is said to have found four or five different proofs himself.) The name of the theorem is misleading--although the theorem is a statement about the algebra of complex numbers, many of the proofs rely upon sophisticated ideas from complex analysis, geometry, and topology.

A high point of this course will be understanding a couple of different proofs of the Fundamental Theorem of Algebra, but the real content will be working through the algebra and geometry of the complex numbers and the theory of analytic functions that are needed to understand the proofs. Students should leave the course with a thorough understanding of complex numbers and functions of a complex variable.

Grades will be based on regularly assigned problem sets that will be a mixture of computational problems and theoretical problems.

There are no specific prerequisites for the course; all the essential topics will be introduced in the course itself, and the course should be accessible to any student with a modest background in mathematics.

Irene Mulvey (B.A., Stonehill College; Ph.D., Wesleyan University) is professor of mathematics at Fairfield University. Her recent publications include "Symbolic Representation for a Class of Unimodal Cycles," Topology and Its Applications (2002), and "Multi-modal Cycles with Linear Map Having Exactly One Fixed Point," International Journal of Mathematics and Mathematical Sciences (2001). Click here for more information about Irene Mulvey.

ENROLLMENT INFORMATION

Consent of Instructor Required: No

 Format: Seminar Level: GLSP Credits: 3 Enrollment Limit: 18

Texts to purchase for this course:
James W. Brown & R.V. Churchill, COMPLEX VARIABLES AND APPLICATIONS

7th Edition (McGraw Hill), Paperback