¿­ÐýÃŹÙÍø

Course Description

Summary

Abstract algebra begins with the algebra of polynomial equations. We all learn (and mostly forget) the solution of quadratic polynomial equations in school, and the "quadratic formula". A corresponding method, and a formula, was discovered in the 1500s for both cubic and quartic equations (involving x to the third or fourth power), but people searched for a method for quintic equations for hundreds of years without success. It turns out that there can be no such general method for quintic equations, and this can be proved. The reason turns out to hinge on a concept of "symmetry" for polynomial equations. This analysis of symmetry turns out to be applicable to physical shapes, molecules, crystals, and tiling patterns. For example, it can be shown that there are exactly 17 essentially different types of tiling patterns possible, and only those. Beyond the analysis of symmetry, abstract algebra analyzes the fundamental operations in mathematics, and so is a powerful tool and unifying concept in many other areas of mathematics and computer science. This class will be historically motivated, building on polynomial algebra and on number theory, and building up to powerful abstract concepts as the term proceeds.