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/s06. There you will find a complete syllabus, course description, current homework assignments, and so on.

Text

Contemporary Abstract Algebra, 6th edition, by Joseph A. Gallian, Houghton Mifflin, 2006.

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!

Exams
We will have two exams during the semester and a final exam.
The final exam will be given Tuesday, May 23, 8:00-10:00, in OHS 319.

Academic Integrity
As a student at Gustavus you are expected to uphold the Honor Code and abide by the Academic Honesty Policy. A copy of the honor code and academic honesty policy can be found in the Academic Bulletin and in the Gustie Guide.

Tests: You are expected to work completely by yourself on tests. I will put the standard honor pledge on the front of each exam for you to sign. The first violation of this policy on an exam will result in a 0 on that exam, and the Dean of the Faculty will be notified, as mandated by the policy. The second such violation will result in failing the course as well as notification of the Dean of the Faculty.

Homework: I encourage you to work on the practice problems and the mastery problems together, but you are expected to work together in an honorable way. 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. I expect that you will write up your work individually and never copy someone else's writeup. You should work on the exam problems on your own. Should I detect students copying each other's work, I will on the first occasion talk with the people having similar work. In case of a second infraction, I will give you a 0 for that assignment and notify the Dean of the Faculty. Any further violation will result in increasing penalties, up to failing the course.

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'', ``mastery problems'' or ``exam problems''.

Practice problems are meant to give you practice solving problems. They will typically be more elementary or computational problems.

Mastery problems are homework problems which are graded on an ``acceptable'' or ``incomplete'' basis. You may turn in any individual mastery 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 mastery grade will be based on what percentage of the problems you finish.

Normally mastery 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, mastery problems will have initial due dates.

Exam problems are problems that each of you must do individually. You are on your honor not to consult 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
It is the policy of Gustavus Adolphus College to provide for the needs of enrolled students who have disabilities. The Advising Center has a Disabilities Services Coordinator to assist you with reasonable accommodation. If you have a learning, psychological, or physical disability for which a reasonable accommodation can be made, you can provide documentation of your disability to the Advising Center (204 Johnson Student Union) or call Laurie Bickett (x7027). It is generally best if this can be done as soon as possible.

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