Fall, 2000

Monday, Tuesday, Thursday, and Friday, 11:00-11:50 p.m.

Buttrick 201

Office Hours. Monday, 10:00-11:00, Tuesday, 10:00-11:00, Thursday, 2:00-3:00. If these hours are not good for you, we can set up an appointment.

Prerequisites. Math 204 (The Art Of Mathematical Thinking) and Math 206 (Linear Algebra).

Text. A First Course in Abstract Algebra, 6^th edition, by John B. Fraleigh.

Course Content. Chapters 1-3, 5, 6, 8, and 9 of the text will be studied. Topics include groups, factor groups, cyclic groups, rings, prime and maximal ideals, fields, extension fields, homomorphisms and isomorphisms, and Galois Theory.

What is "Abstract Algebra"? Well, first we ask, "what is algebra?" Simply put, algebra is about solving equations. In high school algebra, all of the equations have real variables, i.e. variables whose values are assumed to be real numbers. This is rather restrictive, for there are many other very nice number systems out there. In abstract algebra, we try to solve equations where the variables come from different number systems. Much of this class consists of asking, "what other number systems are out there?"

What good is "Abstract Algebra"? At first, it may seem like this class is only an exercise in abstract mathematical thinking. However, there are many applications to what we will be studying. Abstract algebra plays a vital role in:

• Automata: the development of finite-state machines, for example telephones and vending machines.
• Business and Economics: maximizing a set of linear equations constrained by a set of linear inequalities.
• Computer Graphics: 3-D rotations, movement of figures, etc.
• Cryptography: creating and breaking secret codes.
• Information Theory: the design of efficient error-correcting codes.
• Physics: the state of sub-atomic particles.
• Quantum Chemistry: the movements of electrons.

Course Goals. By the end of the semester, you should:

• Understand the notion of abstract algebra

• Be able to write a concise algebra proof

• Be able to read and understand technical mathematics

Class Description. With the exception of the days mentioned below, this will be conducted as a lecture-style class. That is not to say your input won’t be valued or solicited: frequently questions will be posed of you, and (hopefully) questions will be posed of me. If it any time you have a question, please feel free to speak up rather than raise your hand. That’s a power trip I just don’t need.

"Thursday Q&A". Seven days this semester are designated "Thursday Q&A". On these days, we will have time for you to both ask and answer questions. We will take as many questions as possible, and someone will have the opportunity to answer the questions at the board. Of course, you will be rewarded for providing solutions. If you don’t believe me, see "Participation" below.

Attendance. You are expected to attend every class. While attendance during exams and Thursday Q&A will directly affect your grade, missing the other days does not have a direct impact on your grade. It will, however, have an indirect impact on your grade. Trust me. Also, you will not be penalized directly for tardiness, however you are expected to arrive to each class on time.

ecademy.agnesscott.edu. A web page has been set up for this course. You can find it at http://ecademy.agnesscott.edu/Mathematics/mathfacpgs/koch,alan/Math321/index.htm, or if that’s too much typing you can just follow the links from the http://ecademy.agnesscott.edu page. Here you will find all the handouts for the course. Most will be .pdf files, so make sure the computer you’re using has Adobe Acrobat installed.

Here’s how you’ll be graded…

Homework. Each night, there will be homework problems assigned from the sections covered during the lecture. They will consist (primarily) of odd problems from the book, so you can check your answers in the back. This homework will not be collected, but it is assumed that it will be completed by the start of the next class. (Obviously, you won’t be graded on this, but it seemed the most logical place to put this information.)

Participation. You will be graded on participation for this course. Factors that will be used in computation of "participation" include both the asking and answering of questions in class. For Th Q&A, providing solutions to questions will surely help your grade.

Assignments. There will be six assignments that you will turn in. While the questions will tend to be more theoretical than the problems assigned after each section, it is important that you do the daily homework to get a feel for what is going on before you attempt the assignments. Assignments are due by the start of class on the day indicated at the top of the problem set. Late assignments will not be accepted.

You are encouraged to discuss these assignments with the others in the class, but your write-up must be your own. If you have any question about this policy, please let me know.

Honor Code. All students are expected to follow the honor code throughout the semester. Any graded work, be it an assignment, exam, or project, must be pledged (and signed) in order for it to be graded. Please consult the student handbook for more details.

Exams. You will have three in-class exams. The exams will tend to focus more on the calculation-type problems than the theoretical, although this should not be taken as a guarantee. These are tentatively scheduled for September 22, October 27, and November 17. If you have a conflict with any of these dates, let me know ASAP. (The day after the exam is not ASAP.) If you have three exams scheduled for one of these days and wish to take mine at a different time, please let me know as soon as possible so we can make the necessary arrangements.

The exams will cover material from the text, along with material presented in class. You will not be allowed to make-up an exam without a doctor’s note.

Final Exam. The final exam is cumulative, although there will be more emphasis on chapter nine because (a) you were not tested on it during the semester and (b) it encompasses much of the semester anyway.

Project. In addition to the assignments listed above, you must also complete an independent project. The purpose of this project is to

1. give you more exposure to an application of abstract algebra
2. provide you with an easy forum to read a scientific document
3. give you practice in formal mathematical writing

The general procedure for this is as follows. There are eight topics to pick from, each one coming from optional topics in the book (which we as a class opt not to do). Each section is an interesting application of the work done in class. In the interest of it being a true independent project, no two people can do the same project. You must sign up for a section on a first come – first serve basis, with some random phenomenon thrown in to break ties. The project is due on Friday, November 10, although you are encouraged to complete it sooner. It will not be accepted late.

Once a topic is picked, you are to read the section of the text, write a short but thorough description of the results you have read, and complete the indicated exercises. You must show all work – after all, some of the answers are in the back. Here is a list of the topics:

You should not expect a curve to be applied to the point scale, although plusses and minuses will be added as appropriate.