UW Math 497 Spring 2001; Wed 4:30-6:50
Algebra: Solving Polynomial Equations
Department of Mathematics, University of Washington, Seattle, Washington
The principal subject of algebra is the solution of polynomial
equations. The familiar solution of a quadratic or degree two
polynomial equation by the quadratic formula was discovered
independently in several cultures many centuries ago. It is now a
standard part of secondary mathematics education, but typically
students do not study higher degree polynomial equations from the same
perspective. In this course we will do so, as we take a close look at
the most central results in the early history of algebra. These include:
- The solution of quadratic polynomial equations. The quadratic
formula will be examined from three different perspectives.
- The solution of cubic, or degree three, polynomial equations.
This was obtained in the sixteenth century by several Italian
mathematicians and represents the most dramatic advance in algebra to
have taken place for centuries.
- The solution of quartic, or degree four, polynomial
equations. This was also obtained by Italian mathematicians, later in
the sixteenth century.
- The fundamental role of complex numbers in the solution of cubic
equations. Even if one is interested only in real number solutions to
cubic equations with real number coefficients, the method of solution
developed in the sixteenth century led inevitably to the introduction
and study of complex numbers. We will study why this was so and learn
how to use them.
- The attempt to solve polynomial equations of higher degree, culminating
around 1800 with Gauss's proof of the Fundamental Theorem of Algebra.
Underlying these topics is the idea that the coefficients of a
polynomial encode information about that polynomial's roots. Our
goal is to learn how to use the coefficient data to unravel
this hidden information.
Another goal of the course is the development of experience in grappling with
mathematical argument. There will be weekly reading and writing
assignments in which students will be asked to read mathematical
arguments, develop an understanding of the arguments, write out the
arguments in more detail, and write arguments from scratch. Some
class time each week will be dedicated to small group discussions of
these assignments. We will not cover a large amount of material,
but it is intended that an in-depth understanding of this material
will be acquired.
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