Abstract Algebra I
M466, Fall 2012
Lecture: MWF 10:00-10:50, Engineering E205 (code 63370).
Detailed information on homework
Detailed information on project
Sample midterm 2
Extra homework problems
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.
The history of algebra starts with the quadratic formula, first found on Babylonian clay tablets from 1600 BC. Now 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. Our first topic is ring theory, which has applications in many fields such as chemistry, physics, and cryptography. Emmy Noether pioneered a new approach to ring theory in the 1920s, by using ideals to study factorization. In the second part of the course, we will study symmetries of roots of equations. 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.
This perspective relies on 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.
Prerequisite: M366 and M369 or equivalent experience.
Goodman, Algebra: Abstract and Concrete. Chapters 6,7,9.
2) Judson, Abstract Algebra: Theory and Applications. Chapters 14,15,16,18, 19, 20, 21.
3) Garrett, Abstract Algebra. Chapters 3,4,5,6,7,8,9. Click on course notes, or pdf version, or individual chapters.
4) Clark, Elementary Abstract Algebra. Chapters 5,9,11.
5) Ash, Abstract Algebra: The Basic Graduate Year. Chapters 2,3.
6) Wilkons, Abstract Algebra Lecture Notes. Chapters 2,3.
The course grades will be computed as follows:
20% homework; 20% first midterm; 20% second midterm; 5% short project; 35% Final.
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 do an art project or 2 page report. This gives everyone a chance to explore a topic in a new way.
There will be two midterms in class on Friday 9/14 and Friday 10/26. The short project is due Friday 11/30.
The final exam is Monday 12/10, 7:30-9:30 am.
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.
CSU Honor Pledge:
Academic integrity is important to me.
As written by Greg Dickinson, Director of Graduate Studies; Professor, Dept. of Communication Studies:
At minimum, academic integrity means that no one will use another's work as their own. The CSU writing center defines plagiarism this way: Plagiarism is the unauthorized or unacknowledged use of another person's academic or scholarly work. Done on purpose, it is cheating. Done accidentally, it is no less serious. Regardless of how it occurs, plagiarism is a theft of intellectual property and a violation of an ironclad rule demanding "credit be given where credit is due."
Source: (Writing Guides: Understanding Plagiarism. http://writing.colostate.edu/guides/researchsources/understandingplagiarism/plagiarismoverview.cfm. Off-Site Icon Accessed, January 15, 2009)
If you plagiarize in your work you could lose credit for the plagiarized work, fail the assignment, or fail the course. Plagiarism could result in expulsion from the university. Each instance of plagiarism, classroom cheating, and other types of academic dishonesty will be addressed according to the principles published in the CSU General Catalog (see page seven, column two: http://www.catalog.colostate.edu/FrontPDF/1.6POLICIES1112f.pdf).
Academic integrity means having a true educational experience. It involves doing your own reading and studying. It includes regular class attendance, careful consideration of all class materials, and engagement with the class and your fellow students. Academic integrity lies at the core of our common goal: to create an intellectually honest and rigorous community. Because academic integrity, and the personal and social integrity of which academic integrity is an integral part, is so central to our mission as students, teachers, scholars, and citizens, we will ask to you sign the CSU Honor Pledge as part of completing all of our major assignments. While you will not be required to sign the honor pledge, we will ask each of you to write and sign the following statement on your papers and exams:
"I have not given, received, or used any unauthorized assistance."
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 Wed 11-12, Thurs 1-2, or by appointment, in Weber 118.