Math 321 - Foundations of Abstract Algebra

There is no required textbook. I will produce course notes throughout the semester and post them here: Current notes.

If you are looking for other sources to supplement the notes from class, here are some recommendations:

A First Course in Abstract Algebra by John Fraleigh.

Algebra by Michael Artin.

Abstract Algebra by David Dummit and Richard Foote.

Abstract Algebra: Theory and Applications by Thomas Judson. Available Online.

If you would like a book that teaches the fundamentals of reading and writing proofs, I recommend the following:

How to Prove It by Daniel Velleman.

How to Think Like a Mathematician by Kevin Houston.

Mathematical Reasoning: Writing and Proof by Ted Sundstrom. Available Online.

Book of Proof by Richard Hammack. Available Online.

If you would find it helpful to review some of the material from Linear Algebra, please consult my Linear Algebra book.

• Develop the ability to read mathematics with a critical and inquisitive mind, and use these skills to both refine and strengthen your learning process.

• Develop technical writing skills, including the ability to communicate abstract mathematical ideas correctly, clearly, and efficiently.

• Learn how to think axiomatically in order to understand the power, elegance, and beauty of abstraction.

• Learn some important concepts of algebra, including fundamental properties of some algebraic structures (such as groups, rings, and fields), equivalence relations, quotient constructions, and homomorphisms (structure-preserving functions).

Homework assignments will be due on Wednesdays at 5:00pm, and will be posted to the course webpage. You should submit you solutions as one pdf file to the corresponding homework folder within your course OneDrive folder (that I will share with you). Throughout the semester, you may extend the deadline of at most two homework assignments by (up to) two days. If you choose to extend the deadline of an assignment, you should send me an email by the usual due date. Beyond these two extensions, late homework will not be accepted for credit, unless there is an emergency that you bring to my attention before the due date. Your lowest homework score will be dropped.

Although there will certainly be some "computational" problems in the course, most of the homework involves writing proofs and/or detailed explanations. As a result, there are often many correct answers. Moreover, the clarity of exposition and the proper use of mathematical terminology are as vital to your solutions as having the correct idea. A major goal of this course is to learn how to express your mathematical ideas correctly and to write convincing, detailed, and clear proofs. One of my core responsibilities is provide helpful feedback for how to improve your writing. Do not be alarmed if your homework has many comments and suggestions!

One of the most difficult aspects of many people's people mathematical journey is learning how to read mathematics. To get the most out of books and papers, it is essential to work out examples while reading, to constantly ask questions, to isolate what aspects of the material are unclear, and to make conjectures. To help develop these skills, I will give you a few prompts for the assigned reading for Monday and Friday classes. Your responses to the prompts will be due on Gradescope by 10:00am on the corresponding class day. These will be graded for completeness and thoughtfulness, rather than for correctness.

In addition to reading and homework assignments, I expect each of you to engage in the learning process other ways. You should regularly attend class, contribute to class discussions, and ask questions in office hours or via email.

There will be two in-class exams and a scheduled three hour final exam, each of which will focus on conceptual problems and proofs.

In-class exams dates: March 8 and April 28.
Final exam date: Tuesday, May 16 at 9:00am.

If you want to learn how to present your work professionally, as well as keep digital records, I recommend learning how to typeset your solutions. LaTeX is a wonderful free typesetting system which produces high-quality documents at the cost of only a small amount of additional effort (beyond the nontrivial start-up cost of learning the fundamentals). If you plan to do any kind of mathematical or scientific writing in the future, you will likely use LaTeX, so it is worth your time to familiarize yourself with it. See Jim Hefferon's LaTeX for Undergraduates and his LaTeX Cheat Sheet for the basics. Also, feel free to ask me questions about how to use LaTeX, and/or to send you the LaTeX file for homework assignments.

Consult the general Grinnell College policy on Academic Honesty and the associated booklet for general information.

Homework: If you enjoy working in groups, I strongly encourage you to work with others in the class to solve the homework problems. If you do collaborative work or receive help form somebody in the course, you must acknowledge this on the corresponding problem(s). Writing "I worked with Sam on this problem" or "Mary helped me with this problem" suffices. You may ask students outside the course for help, but you need to make sure they understand the academic honesty policies for the course and you need to cite their assistance as well. Failure to acknowledge such collaboration or assistance is a violation of academic honesty.

If you work with others, your homework must be written up independently in your own words. You cannot write a communal solution and all copy it down. You cannot read one person's solution and alter it slightly in notation/exposition. Discussing ideas and/or writing parts of computations together on whiteboards or scratch paper is perfectly fine, but you need to take those ideas and write the problem up on your own. Under no circumstances should you look at another student's completed written work.

I encourage you to look to other books or online sources for additional help in understanding concepts and ideas, but you must cite other books or online sources if they provide you with an idea that helps you solve a problem. However, you may not specifically look for solutions to homework problems, and you may not solicit help for homework problems from online forums.

Exams and Final: You may neither give nor receive help. Books, written notes, computers, phones, and calculators are not permitted at any time during a testing period.

I encourage students with documented disabilities to discuss appropriate accommodations with me. You will also need to have a conversation with, and provide documentation of your disability to, the Coordinator for Disability Resources, located on the ground level floor of Steiner Hall (641-269-3124).

I encourage students who plan to observe holy days that coincide with class meetings or assignment due dates to consult with me as soon as possible so that we may reach a mutual understanding of how you can meet the terms of your religious observance and also the requirements for this course.

• Spend time actively reading the course notes, your notes, and any other sources you choose. I read one chapter of my algebra book nine times during the course of the semester (not counting reading portions of it to help with homework problems). This is normal, and you should not expect to master a section from going to class and reading it once.

• When reading the course notes, do not sit down and read through them like you would a novel. Have a pencil and paper at hand, and work through the arguments and examples on your own. After reading a proof, stop to process it in your own way and then work through it again in your own words. When you see an example illustrating a result, go back through the proof in the context of that one example.

• There will be many new abstract concepts, definitions, and examples introduced throughout the course. In my experience, attempting to memorize them directly is not the most useful way to succeed. Instead, the way to master these ideas is to work with them (both on the homework and on your own) until they because comfortable and natural.

• The material is challenging at both a high abstract level and a detail-oriented technical level. In my opinion, the best way to understand both is to regularly switch back and forth between the two. When trying to understand a difficult theorem, read it several times at different levels (first try to understand the overall structure, then break it down into key points, then work through specific details, and finally repeat the whole process). When working through an example, picture it in the context high level theory we are developing.