# Course Information

Math 31 serves two purposes. The first is suggested by the course title: students should learn the basics of abstract algebra. This means that students should leave Math 31 with a solid understanding of various algebraic structures, specifically structures called groups and rings. In particular, we aim to address the following core questions:

• What is a group? What are some fundamental examples of groups?
• How can we tell when two groups are the same or different?
• How can we analyze the structure of a group?
• What is a ring? How do rings differ from groups?
• How can we use abstract algebra to analyze polynomials and their roots?

The first three questions involve the study of structure and classification of groups. This will require students to become familiar with the concept of isomorphism. The last question leads to the beginnings of Galois theory (which is itself a beautiful branch of algebra). Along the way, we will emphasize the use of groups in the study of symmetries of mathematical objects. This will give a more concrete interpretation of algebra, and it will lend some insight into the origins of group theory.

The second objective of Math 31 is much broader. En route to becoming well-versed in the ideas of modern algebra, students should learn to think like a modern mathematician. In particular, a course like this should help the students learn to read and write abstract mathematics. Both are very much acquired skills, and their development will be nurtured in the course. The most central skill to be developed will be the understanding of mathematical proofs. Students will need to learn how to read proofs, study how they are composed, and become proficient at constructing their own proofs. In addition, the students should learn to convey mathematical ideas clearly and correctly.

The course will progress tentatively as follows. We will begin with a historical overview and some motivating examples, and from there we will begin our study of group theory. This should take roughly the first half of the course, and it will end with the midterm exam. We will then spend two to three weeks on ring theory and polynomials. The course will end with some special topics and possibly applications of algebra, depending on the interests of the class. Class presentations will also take place in the last two weeks of the term. A more detailed schedule will be available on the syllabus portion of the course webpage, and any updates to the course structure will appear there.

Since the reading and writing of proofs is high on our list of objectives, many aspects of the course will be designed to develop these skills. For example, you will be strongly encouraged to read the textbook regularly. Also, the first two or three x-hours will be used as "workshops" for developing proof-writing skills. These will be purely for your benefit, and if you've already taken a proof-intensive class, you need not come regularly. After the first few weeks, we may continue the workshops as needed, or we might begin using the x-hours as problem sessions. Alternatively, we could use the x-hours to cover extra material and/or special topics.

Abstract Algebra: A First Course, Second Edition
by Dan Saracino
Waveland Press
ISBN: 978-1577665366

This is the required textbook for the course. It will be available at Wheelock Books. It is also likely that you can find used copies from other sources, but please make sure that you have the current (second) edition.

In addition to the required text, there will be a supplementary text:

Contemporary Abstract Algebra, Seventh Edition
by Joseph A. Gallian
Cengage Learning
ISBN: 978-0547165097

This book will be placed on reserve at Baker-Berry Library, along with a copy of the third edition. I do not recommend that you purchase a copy. However, it has been used quite often in the past as the textbook for this course, and it may be helpful to consult from time to time.

The course grade will be computed as follows:

 Assignment Number Percent of Grade Midterm 1 30 Final 1 25 Quizzes 8 10 Homework 8 20 Project 1 15

I expect that you will attend class every day. Though it is not officially a part of the course grade, missing class could indirectly affect your performance through homework and exams. In particular, I will not always cover topics from the same perspective as the textbook, and we will even cover some things in class that are not mentioned in the book. Therefore, it would be best to attend class regularly. Additionally, a course like Math 31 is intended to help you build mathematical maturity by learning to read and write proofs, and lectures and in-class discussion will help greatly with this.

On a side note, I will give suggested reading assignments after each class. I highly recommend that you look at them, since they will give you a preview of the next lecture, and they will help you become more comfortable with reading mathematics. I don't expect you to understand every single theorem and its proof on a first reading, but seeing the material ahead of time will likely help you to follow the next day's lecture.