MCS 313 --  Modern Algebra
Fall 2000
Moira McDermott
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

Prerequisites

MCS-220 and MCS-221.

Text

The primary text for the course will be Contemporary Abstract Algebra, 4th edtion, by Joseph A. Gallian, Houghton Mifflin, 1998.

This book is inteded 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 afterwards.  Study the book with a pencil in hand.  Make notes in it.  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!

Evaluation

Your final grade will be assigned using the following percentages as a guide:
 
Homework 25%
Class Work & Participation
5%
Quiz 10%
Exams (2) 40%
Final Exam 20%

Academic Integrity

The academic honesty policy can be found on page 31 of the 2000-2001 college catalogue.  I call your attention to the following excerpt:  "In all academic exercises, examinations, papers, and reports, students shall submit their own work. Footnotes or some other acceptable form of citation must accompany any use of another's words or ideas."

Quizzes and Exams

We will have one quiz, two exams during the semester and a final exam.

The quiz will cover Chapter 0 and is tentatively scheduled for September 19.
The exams are tentatively scheduled for October 10 and November 7.
The final exam will be given December 18,  1:00-3:00,  in OHS 318.

Class Format

A good lecture is usually systematic, complete, precise -- and dull; it is a bad teaching instrument.
                                                                                                          -- Paul Halmos
The best way to learn anything is to discover it by yourself... .  What you have been obliged to discover by yourself leaves a path in your mind which you can use again when the need arises.
                                                                                                          -- George Polya
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.  Class time will be a mixture of lectures, discussions, problem solving and presentation of proofs.  At various times you will be asked to present problems, reflect on the reading and generate questions for your classmates.  It is essential that you come to class prepared to do the day's work.  In particular, you should read the text and attempt homework before coming to 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.

Homework

I hear, and I forget;
I see, and I remember;
I do, and I understand.
                               - Proverb
I encourage you to work with other students on the homework provided that you do so in such a way that every one in your group learns the material.  The most effective way to do this is to first discuss each problem as a group and then have each person work on the problem individually.  When you're done (or stuck) compare your work and discuss it.  Remember that doing the homework is how you learn the material  and that you are not allowed to work cooperatively on tests.

If you do work with other students on the homework, I would like you to follow these guidelines:

  1. Each person should write up the answers independently.
  2. Each person should be able to work each one of the problems independently.
  3. Each person gives credit to the others who helped.
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:

This material can be found in Chapters 1-18, 22, and 24 of Gallian's book.

The second semester continuation (Mathematics 314) may consider fields in more depth and Galois theory, depending on the interests of the participants.

Tentative Syllabus

I expect to cover the following topics.


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