Groups, Rings, and Fields
M466, Fall 2006
Lecture: MWF 9:00-9:50, Engineering E106 (code 344597).
In this course, we will study Galois theory, which is a dramatic and fun subject in the area of groups, rings, and fields. At its heart, Galois theory is about symmetries of roots of equations.
The history of this topic 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 can be studied using field extensions and groups of automorphisms. This will be a highlight of the course.
Prerequisite: M366 and M369 or equivalent experience.
1) Howie, "Fields and Galois Theory, Springer, isbn 1-85233-986-1
2) Reid's course notes on Galois theory.
The course grades will be computed as follows:
25% homework; 25% midterm 1; 25% midterm 2; 25% Capstone project and presentation.
Borderline grades will be decided on the basis of class participation.
Homework: Due every TBA.
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
Detailed information on homework
Capstone project: There are many amazing topics in algebra and Galois theory 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 two midterms in class on Friday 10/6 and Friday 11/17.
The capstone project is due Friday 12/8. There will be no final examination.
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 Monday 2-3 and Wednesday 10-11 or by appointment, in Weber 221.