Abstract Algebra II
M467, Spring 2011
Lecture: MWF 1:00-1:50, Engineering E105 (code 16859).
Detailed information on homework
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 Galois theory and algebraic geometry.
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.
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 cover algebraic geometry, a very powerful branch of mathematics, with recent applications in data analysis and data transfer codes. The highlight here is that geometric properties of curves can be studied using the algebra of their defining equations.
Prerequisite: M466 or equivalent experience. M369 or concurrent registration.
Judson, Abstract Algebra: Theory and Applications (Chapters 18-21).
2) Ash, Abstract Algebra: The Basic Graduate Year (Chapters 6,7,8).
3) Wilkons, Abstract Algebra Lecture Notes (Parts 2,3,4).
4) Reid's course notes on Galois theory.
5) Baker's Introduction to Galois theory.
The course grades will be computed as follows:
10% group presentation; 20% homework; 20% first midterm; 20% second midterm; 30% Final project and presentation.
Borderline grades will be decided on the basis of class participation.
Homework: Due every Friday. Homework is the most important part of this class. Doing homework problems is crucial for doing well in this class. The process of doing homework will help you solve unfamiliar problems on the tests. The homework problems will help you develop skills in algebraic computation and logical reasoning. The grader will only grade homework which is neat, legible, and stapled. I encourage you to brainstorm the problems in groups and write up your solutions independently.
Project: There are many amazing topics and applications of abstract algebra which we will not have time to cover fully. Each person will write a final project and give a presentation on one of these topics. This gives everyone a chance to learn one topic fully and to develop skill at presenting information. In addition, we'll see a snapshot of some fantastic topics.
There will be group presentations in Week 3.
There will be two midterms in class on Friday 3/11 and Friday 4/22.
The written project is due Monday May 9.
Presentations will be during the last week of class and during the final exam period Monday May 9, 2:00-4:00 pm.
There are no makeups for missed exams, regardless of the reason for absence. You must take the final examination at this time scheduled by the university; no final exams will be given earlier. Examinations will not be rescheduled because of travel arrangements. It is your responsibility to schedule travel appropriately.
Help: Help is always available if you have trouble with homework or lecture material. If your classmates can't answer your question, come ask me! Office hours are TBA in Weber 118.