Abstract Algebra I
M466, Fall 2010

Professor: Dr. Rachel Pries
e-mail: pries ATSYMBOL math DOT colostate DOT edu
web page: www.math.colostate.edu/~pries
office: Weber 118, office hours: TBA

Lecture: MWF 10:00-10:50, Engineering E205 (code 63370).

Course description:
Abstract algebra is a fundamental subject in mathematics. Its theory was developed in the 1800s when people realized that many different types of problems could be solved in the same way using underlying algebraic structures. It has applications in many fields such as cryptography and chemistry. In this course, we will study group theory and ring theory.

At its heart, algebra is about symmetries, of geometric objects or of roots of equations. The history of algebra starts with the quadratic formula, first found on Babylonian clay tablets from 1600 BC. During the Renaissance, mathematicians competed to find the roots of cubic and quartic equations. Everyone was stymied by the difficulty of finding roots of quintic (degree five) equations. The first hint of a new approach came when Lagrange showed that the quintic could not be solved by finding functions that are unchanged under permutations of the roots. Finally in 1824, Abel showed that there is no formula for the roots of the typical quintic equation. In modern terms, this is because the alternating group A_5 is simple.

In the second semester, we will study the work of Galois, a French mathematician who lived from 1811-1832. Galois was a political and mathematical revolutionary. His academic career and police record consist of one disaster after another. He died in a dual with a friend over a romantic attachment. In his short tumultuous life, Galois created a new theory, in which polynomials are studied using field extensions and groups of automorphisms.

We will also study elliptic curves, which are curves having a cubic equation. What is special about cubic curves is that there is a group law on the points on its graph. Because of this group law, some points on the graph have special properties. This is an inspiring topic, since elliptic curves played a crucial role in the proof of Fermat's Last Theorem and are used ubiquitously in cryptography.

Prerequisite: M366 and M369 or equivalent experience.