Below, I discuss
Text: Contemporary Abstract Algebra, Fourth Edition, by Gallian.
We all know how to find the roots of a quadratic equation, and generalizations to third and fourth degree polynomials have long been known. It was the quest for a general solution to higher-degree polynomials that lead to the initial development of Abstract Algebra, and one of the more remarkable results of Abstract Algebra is that no such general solution for higher degree polynomials can ever be found!
Abstract Algebra is a mixture of math purely for the challenge and joy of it, and applications (sometimes the applications are to other areas of math, sometimes to areas outside of math). A few applications that have followed from results in Abstract Algebra are: we can not trisect an angle with a straight edge and compass, detecting errors in bar codes, the RSA encryption algorithm, characterizing types of crystals and wallpaper patterns, and solving Rubik's cube
This course will introduce and develop the concepts of groups and rings. The progress of this development will give you further insight into what pure mathematics is all about: building a complex structure from a few (relatively) simple definitions. The goal of Abstract Algebra in particular is to deduce truths about algebraic systems (sets with one or more binary operations) that are independent of the actual elements of the systems, and then to deduce consequences from those truths, thus leading to more truths and more consequences. In this class, hopefully you will begin to understand (if you haven't already) why philosophers and mathematicians have struggled for centuries with the question: Is math created, or is it discovered?
This class will be very challenging, and it will help you develop mathematically. The material is very abstract, and I hope that you find the abstract aspect of it interesting, and even fun. Plan to spend an average of 9 hours a week outside of class working on this course. As usual, some weeks you will spend more time on this class, while others will be less frenetic (relatively speaking)
One of the main points of this course is that you learn how to write rigorous and precise mathematical proofs. While this can be challenging, and you may often have to resort to the old adage ``if at first you don't succeed, try try again'', the process will aid your mathematical development and can significantly improve your clarity of thought outside of math as well.
Because I put so much emphasis on quality proof-writing, you will not be surprised that I'm picky about your problem sets. Your homework should be precise, comprehensible, completely justified, and written in complete sentences. Most of the homework problems will be worth 5 points, and the possible scores will be 5, 4, or No Grade. A longer problem might be worth 10 points, in which case the possible scores will be 10, 9, 8, or No Grade. After I have returned a problem set, you may rewrite any of the problems for me to regrade. Whenever you choose to do a rewrite, you must turn in your rewrite, along with your original paper, within one week of when I return the homework to the class. For example, if I return the homework on Monday, you must turn in your rewrite by classtime the next Monday.
An important aspect of your mathematical development is that you learn to discuss mathematics with others and collaborate on problems. The homework assignments will alternate between Individual assignments and Group assignments. On the group homework assignments, you will work in groups of two and turn in one paper. It is extremely important that both of you understand every solution that your group produces. If one of you has a brainstorm and figures out a tricky problem, the one who had the brainstorm should take the time to make sure the other partner understands it, and the one who did not have the brainstorm should be sure to ask whenever there's a leap of logic they don't follow. On each assignment, one student will be designated as the primary author who writes the solutions, and the role of primary author must alternate between the members of the group. Make a note on your homework of who the primary author was.
For the Individual assignments, I encourage you to work with other students, but each person must turn in a separate paper.
Here are a few guidelines for the presentation of your homework. If you do not follow these, I will return your homework to you ungraded!
|Late problem sets will not be accepted! No exceptions!|
I plan to drop the lowest non-zero problem set score at the end of the semester. If it is warranted, I may be persuaded to drop one zero instead.
You will have two open book, open note takehome exams during the semester. I will give you at least one week to work on each exam. See the syllabus for the due dates.
The final will also be a takehome exam and is due by Noon on Wednesday, December 20.
You will have some quizzes during the semester that will consist of definitions and very basic proofs. I have set aside five dates for these; we'll see whether we actually need five as the semester progresses. I will let you know what your responsibilities are for each quiz. These are intended to be pretty low-stress activities.
I expect to use the weights below, although I reserve the right to change my mind if the semester does not go as expected.
|Two Inclass Presentations||20%|
Although class attendance is not a specified percentage of your grade, I may keep a class roll to help me determine borderline grades at the end of the semester. If you do miss class, you are responsible for the material that was covered.
Please come see me during my office hours! If you have a conflict and cannot make my office hours, please call or email me and we can set up an appointment for another time.
Department of Mathematics and Computer Science
Science Center, Room 109
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278