Math 31: Topics in Algebra
Fall 2022

General Information

Scheduled Lectures

(Section 11) MWF 11:30—12:35
(x-hour) Tu 12:15—1:05
Haldeman Center 028

Instructor

Professor Sarah Frei
Office: 314 Kemeny Hall
Office Hours: Mon 1:00-2:00, Tues 11:00-12:00, Wed 3:00-4:00
Extra Office Hours: Thurs 10/27 12:00-1:00, Fri 11/4 3:30-4:30, Fri 11/11 3:30-4:30
Contact via email: sarah.frei(at)dartmouth.edu (Replace the (at) with an @).

Prerequisites

Math 22 or 24. If you are unsure about your preparation, please talk to the instructor!

Textbook

We will follow the free online text Abstract Algebra: Theory and Applications by Judson, 2022 edition.

A secondary text which is strongly recommended, but not required, is Abstract Algebra: A Student-Friendly Approach by Laura L. Dos Reis and Anthony J. Dos Reis, 1st ed., 2017. See also the errata.

Course Description

This course will provide an introduction to fundamental algebraic structures, and may include significant applications. The majority of the course will consist of an introduction to the basic algebraic structures of groups and rings. Additional work will consist either of the development of further algebraic structures or applications of the previously developed theory to areas such as coding theory or crystallography.

As a result of the variable syllabus, this course may not serve as an adequate prerequisite for Math 81. Students who contemplate taking Math 81 should consider taking Math 71 instead of this course.

Learning Outcomes

By the end of this course, you should be able to:

Weekly homework problems, a midterm exam, a final paper, and final exam will provide students with opportunities to demonstrate the level of their abilities relative to the above learning outcomes.

More generally, mathematics courses should be viewed as an opportunity to develop and strengthen your problem-solving skills; you will be given numerous opportunities to grapple with complex problems and strategize solutions. These are skills you will certainly need to employ throughout as well as after your time at Dartmouth.

Expectations

1. Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.

2. Everyone can have joyful, meaningful, and empowering mathematical experiences.

3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.

4. Every student deserves to be treated with dignity and respect.

*Mathematically, an axiom is a statement accepted as truth, on which all other work is built.

Using these as guiding principles, I expect students in Math 31 to collaborate with me in developing and maintaining an inclusive learning environment where diversity and individual differences are understood, respected, and recognized as a source of strength. Racism, discrimination, harassment, and bullying will not be tolerated. I expect all participants in this course (students and faculty alike) to treat each other with courtesy and respect.

Exams

There will be one midterm exam and a cumulative final exam. The exams are scheduled as follows:

If you have a conflict with the midterm exam because of a religious observance, scheduled extracurricular activity such as a game or performance, scheduled laboratory for another course, or similar commitment, please see me as soon as possible.

Grades

The course grade will be computed as follows:

Homework

You will have weekly written homework assignments. Every week I will assign 5-7 problems whose solutions you can either type up using LaTeX or write up by hand, and then scan and upload to Gradescope. Written assignments will be due on Wednesdays. The first assignment will be due on Sept 21st.

Homework is a very important component of the course; as such, you should expect to spend a lot of time doing homework. An abstract mathematics class is in many ways similar to a language course: you must get lots of hands-on practice to internalize the definitions. Moreover, the homework is a chance for you to develop your mathematical communication skills. To this end, homework will be graded on correctness as well as clarity and exposition.

You may have a 24-hour extension on any three (3) written assignments you choose. You should notify me (via email) of your intention to use the extension before the assignment deadline. No explanation is required. If you believe you need a more significant extension, please email me at least 12 hours before the deadline with your proposed new deadline and a specific plan for how you will complete the assignment while staying current in this class. I will generally approve these requests, unless I notice a pattern of late work, in which case we'll need to have a conversation about necessary adjustments in order to stay better on-top of the material.

A note about LaTeX: LaTeX is a very useful program for typesetting mathematics nicely. If you haven't used LaTeX before and would like to learn, a good place to start is Overleaf}, which is free and provides templates. Alternatively, here is a list of resources compiled by Rebecca Weber on LaTeX. I am also happy to help you get started with this during office hours.

Final Paper

As part of your grade, you will need to write a final paper about a theorem in group theory that we do not cover in class (a list of suggested theorems will be provided). Your paper will contain an introduction with motivation; examples, lemmas, and other preliminary setup; the proof of the main result(s); and a conclusion giving applications or future directions.

The paper will be assigned a few weeks before the end of the term, to be handed in on the last day of class. This will give you ample time to work on the paper. More details will be provided!

Student Office Hours

These are times I have set aside specifically for you. Please come to office hours for any of the following reasons:

You are also welcome to schedule an appointment if you would like to meet outside of the above scheduled times (send me an email).

The Honor Principle

Cooperation on weekly homework is permitted (and encouraged), but you should write up the solution on your own. In other words, share your thoughts (say on a blackboard) to get started, but write up the solution afterwards using only your understanding. Please acknowledge any cooperative work at the end of each assignment.

Plagiarism, collusion, or other violations of the Academic Honor Principle will be referred to the The Committee on Standards. If you have any questions as to whether some action would be acceptable under the Academic Honor Principle, please speak to me beforehand. For more information about standards of conduct, please see these resources.

Student Accessibility and Accommodations

Students requesting disability-related accommodations and services for this course are required to register with Student Accessibility Services (SAS; Getting Started with SAS webpage; student.accessibility.services@dartmouth.edu ; 1-603-646-9900) and to request that an accommodation email be sent to their instructor in advance of the need for an accommodation. Then, students should schedule a follow-up meeting with their instructor to determine relevant details such as what role SAS or its Testing Center may play in accommodation implementation. This process works best for everyone when completed as early in the quarter as possible. If students have questions about whether they are eligible for accommodations or have concerns about the implementation of their accommodations, they should contact the SAS office. All inquiries and discussions will remain confidential.

Mental Health and Wellness

The academic environment is challenging, our terms are intensive, and classes are not the only demanding part of your life. There are a number of resources available to you on campus to support your wellness, including: the Counseling Center which allows you to book triage appointments online, the Student Wellness Center which offers wellness check-ins, and your undergraduate dean. The student-led Dartmouth Student Mental Health Union and their peer support program may be helpful if you would like to speak to a trained fellow student support listener. If you need immediate assistance, please contact the counselor on-call at (603) 646-9442 at any time. Please make me aware of anything that will hinder your success in this course.

Title IX

At Dartmouth, we value integrity, responsibility, and respect for the rights and interests of others, all central to our Principles of Community. We are dedicated to establishing and maintaining a safe and inclusive campus where all have equal access to the educational and employment opportunities Dartmouth offers. We strive to promote an environment of sexual respect, safety, and well-being. In its policies and standards, Dartmouth demonstrates unequivocally that sexual assault, gender-based harassment, domestic violence, dating violence, and stalking are not tolerated in our community.

The Sexual Respect Website at Dartmouth provides a wealth of information on your rights with regard to sexual respect and resources that are available to all in our community.

Please note that, as a faculty member, I am obligated to share disclosures regarding conduct under Title IX with Dartmouth's Title IX Coordinator. Confidential resources are also available, and include licensed medical or counseling professionals (e.g., a licensed psychologist), staff members of organizations recognized as rape crisis centers under state law (such as WISE), and ordained clergy (see https://dartgo.org/titleix_resources).

Should you have any questions, please feel free to contact Dartmouth's Title IX Coordinator or the Deputy Title IX Coordinator for the Guarini School. Their contact information can be found on the sexual respect website at: https://sexual-respect.dartmouth.edu.

Religious Observances

Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with me as soon as possible, or before the end of the second week of the term (at the latest) to discuss appropriate adjustments. Dartmouth has a deep commitment to support students’ religious observances and diverse faith practices.