Overview
Abstract algebra has its roots in the centuries-old efforts of mathematicians to find solutions to polynomial equations.  Over the last few hundred years,  as the subject has evolved, it has become the source of some of the most widely used tools in mathematics. Groups, rings, fields, and related algebraic structures are used in almost all areas of mathematics--analysis, topology, number theory, discrete mathematics--as well as in physics,  chemistry, computer science and other disciplines.  This course will be an intensive study of the basics of "abstract algebra" including the theory of groups, rings and fields.

Course Objectives

• To develop the ability to think abstractly, make conjectures, and understand and construct rigorous mathematical proofs.
• To understand the basic philosophy, purpose and history behind the development of abstract algebraic structures.
• To develop the ability to work with the most common algebraic structures -- groups, rings and fields -- and to become familiar with the most important examples of these structures.
• To understand how mappings can preserve algebraic structure, and through such mappings, learn how to determine when two seemingly different algebraic structures are in fact "isomorphic," i.e., "the same."

Prerequisites
The formal prerequisites are MCS-221 and MCS-220 or MCS236. More to the point, you should be comfortable with creating and writing proofs, basic logic, and elementary set theory. A familiarity with matrices and determinants is also helpful.

Course web site: The best source of information about this course is available at www.gac.edu/~mmcdermo/mcs313/f02. There you will find a complete syllabus, course description, current homework assignments, and so on.

Text

Contemporary Abstract Algebra, 5th edition, by Joseph A. Gallian, Houghton Mifflin, 2002.

This book is intended to be read.  For each class session, you are encouraged to read the pertinent portion of the text at least once  beforehand and at least twice afterwords.  Study the book with a pencil in hand.  Make notes in it.  Gallian is sometimes terse. Even exercises with "answers in the back" often leave you wondering "huh"? Fill in those blanks when you are reading. Mark where you have questions.   Do NOT try the exercises without reading the text;  simply skimming the examples is not sufficient.   You will find that it will be  necessary to read the text several times before attempting any exercises.  To survive this course, you must learn how to read a math book!

Quizzes and Exams
We will have one quiz and two exams during the semester and a final exam.
The final exam will be given Wednesday, December 18,  1:00-3:00,  in OHS 318.

Academic Integrity
You are expected to work together in an honorable way in this course. This means that while you can discuss problems and their solutions, each of you should make a real effort to solve each problem by yourself, and you should give credit to any people or texts that helped you find solutions. Needless to say, you are expected to work completely by yourself on tests and honor problems.

Cheating is not allowed in this course. If I find someone has cheated, then I will take action ranging from flunking the assignment in question to flunking the entire course. I also notify the Dean of Students.

The academic honesty policy can be found in the 2002-2003 college catalog.

Classes
We learn by thinking and doing, not by watching and listening.  Learning is an active process:  it is something we must do, not have done to us.  Classes will be used for lectures, problem solving, discussions, and other fun activities. You should prepare for classes by doing the reading beforehand (reading assignments are posted on the Web),  thinking about the problems in the text, and formulating questions of your own.  You should also participate as much as possible in class.  Class meetings are not intended to be a complete encapsulation of the course material.  You will be responsible for learning some of the material on your own.

Attendance, both physical and mental, is required.

Should you need to miss a class for any reason, you are still responsible for the material covered in that class. This means that you will need to make sure that you understand the reading for that day, that you should ask a friend for the notes from that day, and make sure that you understand what was covered. If there is an assignment due that day, you should be sure to have a friend hand it in or put it in my departmental mailbox (in Olin 324). You do not need to tell me why you missed a class unless there is a compelling reason for me to know.

Homework

I hear, and I forget;
I see, and I remember;
I do, and I understand.
                               - Proverb

I will assign homework at the beginning of each chapter by posting them on the web. The problems will be designated as ``practice problems'', ``portfolio problems'' or ``honor problems''.

Practice problems are meant to give you practice solving, writing and reading mathematics. For these problems, you will be assigned a group or partner. You should attempt all of the problems and then arrange to meet with your group and compare solutions. Once a week you will write a brief a summary of your meetings telling me what difficulties you had and what problems you were unable to solve. These problems will not be handed in, nor will they count in your final grade. However, this kind of practice will greatly help your performance on the other homework and on the tests.

Portfolio problems are homework problems which are graded on an ``acceptable'' or ``incomplete'' basis. You may turn in any individual portfolio problem whenever you think you have it solved. I will return it to you as quickly as I can, but normally with only an indication of whether it is acceptable or needs more work. (Sometimes I may give a brief indication of what area it needs more work in.) If a problem needs more work, and you aren't sure what sort of work it still needs, you should treat that as an invitation to come talk with me about it. Once you've done the additional work, you may turn the problem in again, attached to (or clearly marked on) the original. In fact, you may turn each problem in as many times as you like, until it is marked as acceptable.

You may discuss the problems with each other, if you like, but I recommend that you work more independently, since  you will learn more that way.  Also, you should write up your solution independently even if you discuss the solution with another student. Remember that doing the homework is how you learn the material  and that you are not allowed to work cooperatively on tests. Your portfolio grade will be based on what percentage of the problems you finish.

Normally portfolio problems may be turned in at any time up until the day before the exam covering that material. However, if we would benefit from discussing a problem in class, I may issue a "last call" for solutions to that problem, at least a week in advance. Also, in order to keep us on track, especially in the beginning, portfolio problems will have initial due dates.

Honor problems are problems that each of you must do individually.  You can think of these as miniature take-home tests; you are on your honor not to cheat by consulting other people or books.  These problems will be graded in the usual way, and, except in extreme circumstances, you only have one chance to do each one.

Course Material
Algebra is one of my favorite subjects and is filled with powerful ideas and beautiful proofs. The course can be loosely divided into two segments:

• Groups:  examples and properties
• Rings, integral domains, and fields: an introduction

I expect to cover the following topics.

• Preliminaries:  Chapter 0
• Groups:  Chapters 1-11
• Rings:  Chapters 12-18

Evaluation
Your final grade will be assigned using the following percentages as a guide:

Accessibility:  Please contact me during the first week of class if you have specific physical, psychiatric, or learning disabilities and require accommodations. I will do my best to facilitate the  necessary arrangements.  All discussions will remain confidential.

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