Introduction to Abstract Algebra I MATH 321 Fall 2025

A traditional first course in abstract algebra typically covers an axiomatic approach to algebraic structures, including groups, rings, and fields. While these are exactly the sorts of mathematical objects that mathematicians study when they study algebra as an area of mathematics, such a presentation of the subject can feel very disconnected from what most people think of as algebra, as it is taught in high school.

One of the goals for this course is to understand why algebra as it is seen in high school has evolved into the rich and abstract subject of modern mathematics that it is. We will start by trying to understand basic algebraic tasks, like solving equations and factoring polynomials, but at a deep level. This will necessitate an investigation of the structures of the numbers and operations involved. We will define types of structures including fields, rings, and groups.

Then, with these new and beautiful structures at our fingertips, we will have no choice but to explore them in their own right. What can we learn about their nature? While we uncover their mysteries, these algebraic structures will help us grow as mathematicians and teach us to think and justify our reasoning in new ways.

But we will try not to lose ourselves in the abstract (that’s what a second course is for). How can these structures help us resolve mysteries about the algebraic tasks we started with? Let’s explore that together.

While we will learn to do more than just the following tasks, these are the key learning targets for the course, organized into four categories.

You will have multiple opportunities to demonstrate your mastery of the learning targets. Your first opportunity will be during weekly learning target quizzes, usually given at the end of class on Fridays. Each quiz will be clearly aligned with one of the learning targets. Quizzes will be graded and assigned a grade of one of the following.

Quizzes will be graded and assigned a grade of U, P, or M, indicating you have demonstrated Unfamiliarity, Partial Understanding, or Mastery of the learning target.

For any learning target for which you do not demonstrate Mastery on your first attempt, you may reattempt to do so by taking another learning target quiz within two weeks of the original quiz. Additional attempts, and attempts beyond the two week window, will be granted only after meeting with me to discuss your progress and understanding of the learning target.

Occasionally, a grade of M- may be assigned if the quiz contains minor mistakes that could easily be corrected. In such cases, you may be able to elevate your grade to M without taking an additional quiz by revising your quiz appropriately (usually through discussion with me).

Your grade in the course will be partially determined by the number of learning targets for which you have demonstrated Mastery. There is no credit given for learning targets below Mastery level, and the number of attempts it takes to reach Mastery does not matter.

The final exam for this class is on Monday, December 8 from 1:30pm to 4pm. Instead of a traditional final exam, we will use this time as an opportunity for you to complete any learning targets you have not yet mastered.

In the unlikely event that the University is closed due to inclement weather on the day of our exam, we will hold the exam at the same time on Friday, December 12 (the designated makeup day for this situation). In the event that this makeup day is also canceled, I will contact you with alternative options.

Your final grade will be calculated using the following table:

To receive a particular grade, you must meet or exceed the requirements for that grade in both categories. If you do not meet the requirements for a D, your final grade will be an F.

It is also possible to receive a plus or minus grade if your engagement score is at a level above or at most one level below the level specified for the number of learning targets mastered. For example, if you have mastered 11 learning targets and have over 90% for your engagement score, your final grade would be a B+. If your engagement score is more than one level below your mastery score, the two levels will be averaged.

All your grades for all assignments will be kept inside Canvas, so you can always check to see how you are doing in this class.

However, note that your final grade depends on two measures: your mastery of the learning targets and your engagement score. The “Total” you see in Canvas is the total for the engagement score only. There is also a Learning Mastery tab where you can see which learning targets you have mastered.

You are expected to attend every class period. It is impossible to receive participation points if you are not in class.

But what happens if you absolutely cannot attend class? Maybe you are very ill, somebody has slashed your tires, and you are having a bad hair day! These things happen, and I will try to be understanding when you communicate with me about your situation. Please reach out to me as soon as possible to discuss your options.

One important note though: in this class, just like in every other class, missing a day of class, even with a perfectly valid excuse, does not excuse you from learning what was covered that day. As your instructor, I will try to help you catch up, but this will take more work than it would have if you were in class.

Late work will be accepted only under extenuating circumstances and must be discussed with me in advance. Deadlines are set to encourage you to stay on track with the course material and manage your time effectively. If you are struggling to meet a deadline, please reach out to me as soon as possible to discuss your options.

Turn off your cell phones when in class and keep them put away, arrive on time, and do not pack up your things before the end of class. All of these are potential distractions to yourself and your classmates that could negatively impact your learning.

Additionally, we expect that you will treat all members of our course community with respect and kindness. In doing mathematics, or almost anything worth doing in life, you are going to make many errors and false starts while becoming more proficient. Think, for example, of learning to play a musical instrument, or learning an athletic skill, or developing a friendship. We want to establish a classroom atmosphere where the inevitable struggles and mistakes become an opportunity to learn and grow; not an opportunity for embarrassment. Thus, please be constructive and polite in questioning your colleagues in class.

Don’t cheat! It is expected that members of this class will observe strict policies of academic honesty in every aspect of this course. In particular, you are expected to solve homework problems by yourself or together with your group, and not find solutions online. In general, UNC’s policies and recommendations for academic misconduct will be followed. For additional information, please see the Student Code of Conduct at the Dean of Student’s website http://www.unco.edu/dos/Conduct/codeofconduct.html.

The use of generative AI tools (such as ChatGPT, Google Gemini, or CoPilot) is allowed in this course, but it is your responsibility to ensure that you use of these tools with academic integrity.

What does this mean? Ask yourself what you hope to get out of this course. I really hope your answer involves learning new skills, becoming a better thinker, and understanding mathematics at a deeper level (even if these goals are in service to eventually getting a better paying and more fulfilling job).

Warning: As good as AI has become recently, it is still not great at mathematics, so anything you get out of a generative AI tool should be approached with a fair amount of skepticism. This is also a good way to learn mathematics though: play “find the mistake” in what AI gives you!

To be clear, you should NOT use AI tools to shortcut the learning process. Getting someone else to do your work for you (even if that someone is a robot) is plagiarism and a violation of the academic integrity policy.

The University of Northern Colorado is committed to providing a safe and inclusive learning environment for all students that is free from discrimination and harassment, including sexual harassment, sexual assault, domestic violence, dating violence, and stalking. Students who have experienced (or who know someone who has experienced) any of these concerns should know that they are not alone. UNC has staff members in the University’s Office of Institutional Equity and Compliance (OIEC) who are trained to support students in navigating these concerns and are able to provide on- and off-campus resources and supports, referrals to health and counseling services, academic and housing modifications, and mutual no-contact orders between individuals.

Please be aware all UNC instructors and most staff members are required to report their awareness of sexual misconduct to the OIEC. This means that if students tell an instructor about a situation involving sexual harassment, sexual assault, dating violence, domestic violence, or stalking, the instructor must share that information with the Title IX Coordinator and Equity Officer, Jimmy Kohles. Mr. Kohles or a trained staff member in OIEC will contact the reporting students to let them know about resources and support services at UNC as well as their options to pursue an investigation through OIEC, law enforcement, or both. Students who have experienced these types of incidents are not required to speak with OIEC staff regarding the incident. Students’ participation in OIEC processes are entirely voluntary.

If students do not want the Title IX Coordinator notified, instead of disclosing this information to the instructor, students can speak confidentially with the following people on campus and in the community. They can connect you with support services and help explore options now, or in the future. UNC has confidential victim advocates available 24/7 by phone - students can contact the Assault Survivors Advocacy Program (ASAP) at 970-351-1490 to seek confidential guidance and support.

My goal is to co-create a learning space that is inclusive and responsive to your access needs. If you anticipate or experience barriers to full participation in this course for any reason, please connect with me. Whether it’s related to course design, content delivery, or materials, I welcome that dialogue.

If you are connected with the Disability Resource Center (DRC), please share your accommodation letter early so we can plan together.

If you are not yet connected but would like to explore accommodations or access resources, you can contact the Disability Resource Center: 970-351-2289, [email protected], Michener L-80, www.unco.edu/disability-resource-center.

Significant numbers of UNC students are challenged in getting enough food to eat. Others have difficulty finding safe and stable housing or meeting other essential needs, such as medical and mental healthcare, childcare and transportation. Still others face financial crises in the wake of unexpected emergencies.

Through its Center for Student Well-Being (CSW, located on the lower floor of the UC), UNC offers personalized 1-on-1 assistance to students facing food insecurity, housing insecurity, financial emergencies or other basic needs challenges. UNC also provides students with access to several food resource programs on campus, including the Bear Pantry, where students can access food and hygiene items at no cost. Resource navigators in the CSW can also help students gain access to the Supplemental Nutrition Assistance Program (SNAP) and other resources.

If you’re not sure where to turn, the website for UNC’s Student Outreach and Case Management office lists a wide variety of resources for students. Case Managers can assist students during difficult circumstances which may include medical, mental health, personal or family crisis, and illness or injury.

Mental health professionals are available on-campus and in the community. See a wide variety of on- and off-campus resources on the Dean of Student’s webpage.

For free, confidential consultations, check out the Counseling Center. To access staff in the Counseling Center, call 970-351-2496 or stop by the Center, located on the second floor of Cassidy Hall.

If you or someone else is experiencing a crisis or suicidal thoughts and the Counseling Center is not available (including evenings, weekends, and holidays), contact either North Range Behavioral Health by calling 844-493-TALK (8255) or text TALK to 38255 or the 988 Suicide and Crisis Lifeline at 988.