Welcome to Topics in Algebra! Algebra is one of the fundamental areas of advanced mathematics. This may be surprising to you, because you have been doing algebra for years! While algebra started as the science of solving equations, roughly 200 years ago it grew to encompass the study of "algebraic structures." In this class, we will learn the basics of some of these algebraic structures, how to work with them, and see some of the power of modern algebra in terms of applications.

This class will be run as a mix of asynchronous and synchronous activities. Lectures will be pre-recorded, and posted here. Office hours and problem-solving sessions will be synchronous, and hopefully happening at a wide enough range of times that everyone has the ability to participate.

The remote nature of this class, and the desire to have a record of such for asynchronous learning will require us to record some learning that happens during office hours and problem sessions. As such, familiarize yourself with the Statement concerning recording of classes.

Find out more course information at the Syllabus page, and what you need for day-to-day class at the Daily Materials page.

Course Summary

This course is an introduction to the study of algebraic structures. We will keep the occasional abstractness grounded in plenty of examples. While requiring some definitions to get there, we will end up studying some familiar objects like the integers and n-by-n matrices. What we learn from the properties of these examples can lead to incredibly powerful results about more general algebraic structures. With the fundamentals in hand, we will turn to some applications of algebra in the modern world.

I have three major goal areas for you in this course, which are:

Content: know the definitions and properties of algebra fundamentals, groups, and rings
Proofs: know common methods of proof (induction, construction, contradiction, contrapositive), and be able to write clear, concise proofs
Engagement: find an area where algebra intersects your life, and think about how the two fit together

Goals should be measurable, so I will do my best to measure all three of these goals. We will have daily and weekly homework, exams, and a final project. We will also have other graded pieces of work, including surveys, exam wrappers, and feedback.

Learning is an active process, which requires your participation. With this in mind, I will do my best to present all the content to you, but a large chunk of learning will happen not when I lecture, but when you ask questions. I want you to be successful in this course, and will do everything I can to help you work through your growing understanding of algebra.

The above includes structural help also. If you have conflicts or needs that I can help you work through, let me know. These include applying disability-related academic adjustments and services to the context of the course; the earlier you let me know about any such needs, the better, but whenever is most comfortable for you is good for me. Conflicts, in the form of religious observances, athletics, or life external to this class will be met with all possible accommodations.

The academic environment at Dartmouth is challenging, our terms are intensive, and classes are not the only demanding part of your life. This is even more true in this time of remote learning, and uncertainty in the world. There are a number of resources available to you to support your wellness, including your undergraduate dean, Counseling and Human Development, and the Student Wellness Center.