MATH 330: Abstract Algebra

"It looked absolutely impossible. But it so happens that you go on worrying away at
a problem in science and it seems to get tired, and lies down and lets you catch it."
- William Lawrence Bragg, won the Nobel Prize at age 24.

Textbook:

Contemporary Abstract Algebra , 8th Edition, by Joseph Gallian.

Please note that we will work on developing your independent reading skills in Mathematics and your ability to learn and use definitions and theorems. I certainly won't be able to cover in class all the material you will be required to learn. As a result, you will be expected to do a lot of reading. The reading assignments will be on topics to be discussed in the following lecture to enable you to ask focused questions in the class and to better understand the material.


Course Description:

Topics covered: Topics are subject to change depending on the progress of the class, and various topics may be skipped due to time constraints.

Welcome to our course in Abstract Algebra. By the time you take this course, you should be at the point where you are fairly comfortable with various techniques of proof. We will be using and perfecting these techniques and introducing new methods as necessary. BE WARNED: If this is your first proof-based 300-level math course, then this class is probably not for you. This course requires a certain amount of mathematical sophistication. You are expected to work hard and write quality, thoughtful proofs. You will be asked to write rigorous arguments in order to solve the problems. We will be doing a significant number of proofs, and everyone should ALREADY be fairly comfortable with the process. By the end of this semester, you will be mathematical geniuses!

The basic premise of abstract algebra, or modern algebra, is to extract the basic ideas and properties of algebra as you know it from high school algebra and linear algebra and abstract them to be used in more general ways. In previous algebra classes we tend to study a certain set of objects, such as numbers or functions or matrices, and then we learn about their algebraic properties. In abstract algebra, we begin with the basic properties and structure, without much emphasis on the objects they are applied to. This means we will study algebraic properties apart from concrete realities, specific objects, or actual instances. This is theoretical mathematics at its finest!

Many of the topics we will learn in this abstract algebra course can be applied within other areas of mathematics, including algebraic topology (my field of expertise,) algebraic geometry, algebraic number theory, and others. The notions of abstract algebra can also be applied in many other scientific realms, including physics, chemistry, coding theory, quantum mechanics, neuroscience, anthropology, and many others. While we won't be exploring these applications much, if at all, in this course, it is important to realize that just because something is studied from an abstract, theoretical point of view, that doesn't mean that it is not useful. There are wide ranging applications throughout all of science and engineering and mathematics.

Upon successful completion of this course, students will be able to:

  • Assess properties implied by the definitions of a group and rings,
  • Use various canonical types of groups (including cyclic groups and groups of permutations) and canonical types of rings (including polynomial rings and modular rings),
  • Analyze and demonstrate examples of subgroups, normal subgroups and quotient groups,
  • Analyze and demonstrate examples of ideals and quotient rings,
  • Use the concepts of isomorphism and homomorphism for groups and rings, and
  • Produce rigorous proofs of propositions arising in the context of abstract algebra.


Grading:

Your overall grade for the course will reflect how well you are doing and will be high if you are working hard on the homework and doing well on the exams. Many of the questions on the exams will be in the same spirit as the homework questions. Therefore understanding how to do all the homework questions will enable you to do well on the exams.

In addition to learning wonderful mathematics this semester, we will also be working on three academic fundamentals that are vital to success in your education and in your career: Reading, writing, and communication.

  1. Reading the textbook is extremely important. It is required, NOT optional. I certainly won't be able to cover in class all the material you will be required to learn. As a result, you will be expected to do a lot of reading. The reading assignments will be on topics to be discussed in the following lecture to enable you to ask focused questions in the class and to better understand the material. Your chances of getting a good grade in this course are infinitesimally small unless you read the textbook in addition to attending class.
  2. Writing quality proofs is essential. You are expected to be able to effectively present mathematics with a well-organized, thoughtful, neatly written argument. What is a "quality proof"?
  3. Communication with your peers will be a focal point. In addition to working with partners on your homework assignments, there will also be regular discussion and presentations given in class. Students will be asked to present their solutions to problems to the rest of the class. The best way to truly understand a concept is to be able to explain it to someone else. Talking about math is just as important as thinking and writing math. Our goal is to become more comfortable with the necessary communication.

"I hear and I forget. I see and I remember. I do and I understand."
- Chinese proverb

"You can't be a passive learner in a class like this. You can't zone out. You have to come prepared for class because there will be participation, and you'll have to demonstrate you know the material."
- Student


Homework: There will be regular homework assignments which must be turned in by 5:00 pm on the due date. Pick a homework partner that you will be working with for the semester. (Depending on how the class does, I may have you switch partners during the semester.) Each student should work out every problem, but you and your partner will only turn in one combined final version of each assignment. Please note that you are welcome to work in groups of as many people as you want, but you and your partner are responsible for writing up and handing in your own pair's homework. Each pair of students should submit their own work, not a handwritten copy of someone else's. Also, you should feel free to write out your assignments by hand, but I encourage you to typeset your homework using LaTeX. Each individual student is STRONGLY ENCOURAGED to think about and attempt the problems on their own before meeting with their partner. When you are about to tackle the homework assignment, you should use this strategy: THINK, PAIR, SHARE. When working in groups, please be careful that you are actively participating in the process. Please be careful that you are able to work all of the problems on your own before the exam time arrives, with no coaching from a friend. Please use whatever resources aid you in learning the material, including computer assistance, office hours, other students, professors, other math books, etc. However, plagiarism will NOT be tolerated, and this includes copying or paraphrasing someone else's work.

Exams: There will be two Midterm Exams and one Final Exam. Exams are closed book, closed notes, closed friends, and open brain. Use of calculators, phones, and other electronic devices will NOT be permitted during exams. The dates of the exams will be decided a week or two in advance.

Class Participation: Class participation will be based on your willingness to ASK and ANSWER questions in class. There will be active discussion at certain times, and you will also be required to present some proofs to the class. It is imperative that you keep up with the reading assignments and work some of the discussion problems before you come to class. This will help you answer my questions and help you ask more essential, thought-provoking questions during class time.


Extra Help:

It is essential not to fall behind because each lecture is based on previous work. If you have trouble with some material, SEEK HELP IMMEDIATELY in the following ways:

  • ASK ME! (either in class or privately),
  • One of the very best resources may be your fellow students!

If you are having any difficulties, seek help immediately - don't wait until it is too late to recover from falling behind or failing to understand a concept!

Accommodations: SUNY Geneseo is dedicated to providing an equitable and inclusive educational experience for all students. The Office of Accessibility (OAS) will coordinate reasonable accommodations for persons with disabilities to ensure equal access to academic programs, activities, and services at Geneseo.

Students with approved accommodations may submit a semester request to renew their academic accommodations. More information on the process for requesting academic accommodations is on the OAS website.

If you have questions, contact the OAS by email, phone, or in-person:

Office of Accessibility Services - Erwin Hall 22, (585)245-5112, access@geneseo.edu.