The College of the Holy Cross

Mathematics 43, section 1 -- Algebraic Structures

Syllabus, Fall 1998

Professor: John Little
Office: Swords 335
Office Phone: 793-2274
email: little@math.holycross.edu or jlittle@holycross.edu
Office Hours: MWF 10-12, TR 1-3, and by appointment.

Preparation for Advanced Mathematics

Algebraic Structures is the first half of the year-long second year algebra sequence designed primarily for Mathematics majors. The continuing course in the spring is called Linear Algebra. These courses have two major goals:

To introduce the foundations of the branch of mathematics known as modern or abstract algebra, including linear algebra, and
To develop students' familiarity with and skill in applying the basic strategies for developing and basic techniques for writing mathematical proofs.

Within the Mathematics major, these courses (together with the Principles of Analysis course) serve as bridges from the basic mathematics you have seen in high school and in the Analysis 1,2,3 courses to the more advanced courses you will normally take in your junior and senior years.

Because of this, Algebraic Structures and Linear Algebra can perhaps best be described as ``boot camp for Math majors''. Both of the goals 1. and 2. above come into play here. More advanced courses will draw regularly on the concepts introduced here (algebraic structures such as groups, fields, vector spaces, linear mappings and matrices, etc.) and in those courses, the professors will assume that you are familiar with the properties of those structures that we will prove in this course. In addition, you will be expected to be able to develop and clearly present logical proofs of your assertions in those courses, so they will also assume the basic techniques and strategies we will discuss here.

Make no mistake, there will almost certainly be times when the ``boot camp'' analogy seems painfully apt -- many Mathematics majors find these courses to be among the most challenging they take at Holy Cross because the whole way we work in this course may well seem unfamiliar and baffling at first:

Many problems we will do will not have a single number or formula as an answer and more generally, learning ways to perform calculations will almost never be the main focus
Even the calculations we do will rarely involve particular explicit numbers, functions, etc.
You will definitely not master this material by learning rote procedures for solving a set collection of different types of problems
A plausible reason for expecting something is true will not be sufficient.

Instead, in many problems the goal will be first to develop, and then to explain clearly, the complete logical argument that establishes the truth of a general statement, starting from a clearly stated set of assumptions and using results established previously. You will have to think about the logical structure of the subject matter and understand the definitions of concepts and the statements and proofs of theorems to get the ``skeleton'' of the subject firmly in place in your mind. You will need to commit a collection of key examples to memory and be able to reason about their properties. You will need to document much more of your thinking about problems than you probably have done before. While a good intuition may guide you to correct statements, just making correct statements will not be enough.

At times, you may ask why we are doing this at all. The ultimate answer is that this ``abstract'' proof-oriented work is the way all mathematics is communicated and in a sense it is what most of advanced mathematics is really about, so you have to be prepared for it if you decide to continue! The concept of mathematical proof is the unique and distinctive feature of this branch of knowledge; I think it is no exaggeration to say that it is one of the crowning achievements of the human intellect. Even applied mathematicians (ones who work on problems directly inspired by problems in the real world) must develop new ideas to solve those problems, and then provide convincing evidence (proofs) that what they claim is true so that others can follow what they do. Lest this sound too grim, be aware that you will have lots of chances to develop and practice these new skills, and I will always be willing to give you the benefit of my experience working with this kind of mathematics! Even if you find this difficult at first, persistence and openness to a different way of thinking will usually pay off in time.

Text

The text book for the course is Elements of Modern Algebra by J. Gilbert and L. Gilbert, 4th edition. We will cover most of the material in Chapters 1 - 4 and the Appendix on logic this semester.

Material We Will Study

During the semester we study the following topics

Unit I: Logic and set theory (about 9 days)
Unit II: The integers and the integers modulo n (about 9 days)
Unit III: Algebraic structures -- groups rings and fields (about 9 days)
Unit IV: First steps in the general theory of groups (about 9 days)

As always, it may become necessary to add, subtract, or rearrange topics. I will announce any changes in class and on the course homepage.

Course Format

In order for a student to get as much as possible out of this or any course, regular active participation and engagement with the ideas we discuss are necessary. To get you more directly involved in the subject matter of this course, regularly throughout the semester the class will break down into groups of 3 or 4 students for one or more days, and each group will work individually for (a portion of those) class periods on a group discussion exercise. I will be responsible for designing and preparing these exercises, and I will be available for questions and other help during these periods. Each group will keep a written record of their observations, results, questions, etc. which will be handed in. The other meetings of the class will be structured as lecture/discussions.

Grading Policy

Grading for the course will be based on

Three in-class tests, together worth 45% of the course grade. Tentative dates:
- Friday, September 25.
- Friday, October 23.
- Friday, November 20.
A three-hour final exam, worth 30% of the course grade. The final examination will be given at 2:30 p.m. on Monday, December 14.
Written reports from small group discussions -- one report from each group. Information regarding the expected format will be given out with the first assignment of this kind. Together, worth 10% of the course grade.
Individual homework assignments, given out in class. The homework will count as 15% of your course grade. The individual assignments are a very important part of this course and keeping up to date will be necessary to succeed with this material. No credit will be given for late homework, except in the case of an excused absence, or with my permission.

If you ever have a question about the grading policy or your standing in the course, don't hesitate to ask me.