Syllabus Math 457: Abstract Algebra II

Recommended Texts: The following are all excellent resources.  I highly recommend you have at least one of these for your personal copy and then refer to the hearth copies I will provide outside my office door for other sources. I am NOT suggesting you buy a copy of each of these.  Pick the one that sounds best for your needs and learning style.

•  Abstract Algebra by Dummit and Foote  - advanced but thorough and a good resource for anyone going to graduate school in mathematics.  Proofs and material are presented in a very dense and at times hard to read format, but they are largely complete.

• Contemporary Abstract Algebra, (4th ed or higher), by Gallian, accessible and easy to read.  Proofs given are often not complete, but all the stepping stones are provided and filling in the missing pieces is a good exercise

• Abstract Algebra: An Introduction by Hungerford - Similar to Contemporary Abstract Algebra.  Maybe a ever so slightly less accessible/readable but proofs are a little more complete.  Approaches rings first and then fields which can be a little disorienting if you are learning in the opposite order, but makes a lot of sense from a Number Theoretic stance.

• a Book of Abstract Algebra by Pinter - extremely easy to read, very brief but effective coverage of material.  Skips over a lot of the content to get to the heart of things.  Details are often filled in with exercises at the back of the chapter so you need to at least read all the exercises to cover the content.

To paraphrase Stefan Banach, a mathematician is someone who sees analogies - analogies between theorems, between proofs, between theories, an ultimately between analogies themselves. 

Class Objectives / Big Questions

Is there an abstraction of the ideas running through a number of disparate areas of mathematics which, when accurately defined, gives greater insight into these individual areas? Does this abstraction allow for discoveries which apply to the whole without consideration of the characteristics which distinguish the parts? By reaching the level of abstraction which Abstract Algebra provides, we will explore some of the most beautiful and unifying results in mathematics. In this course you will:

• learn the fundamentals of rings, ideals, and fields;

• continue to develop your mathematical maturity and independence by asking good questions, forming conjectures, independently learning material;

• combine creative and algebraic approaches to problem-solving, while identify underlying abstract patterns;

• hone your stellar proof-writing skills; and

• participate in the mathematical community.

Individual Exams:  There will be two or three individual exams (celebrations? denouements? an assessment by any other name is just as exciting).  These will focus on your knowledge of definitions and theorems, your big picture understanding of the course content, and your ability to apply concepts to straightforward calculations 

Weekly Quizzes:  There will be a short quiz most weeks consisting of one proof (similar to something from that week's homework assignment) and possibly a definition or theorem statement that would have been used in that week's homework.  You will be allowed a notecard consisting of definitions and statements of theorems, but no proofs, for these quizzes. 

Problem Assignments & Group Discussions: Throughout the semester I will assign homework sets that reinforce class concepts and provide the opportunity to apply the theorems and techniques discussed in class.  These homework sets should be done collaboratively with your assigned homework group.  You can either meet to do the problems together, or work on the problems individually and then meet to go over your solutions and select which solutions to turn in.  You will turn in a specified number of LaTeX-ed solutions and a 5-15 minute recording of your group conversation.  For the last third of the class, we may switch to a different form of homework consisting of more in-class presentations.  

Engaging in the Mathematics Community:  Students are expected to attend 4 of the math colloquium talks given throughout the semester. A schedule of talks can be found on the Math Department website, and linked to from the course WISE site.  The goal of this attendance requirement is to introduce you to the diversity and vitality of current mathematics research, and to include you in the Math Department community.  In addition, there will be in-class investigations. Your grade for this component of the course will be based on how meaningfully you engage in these activities and discussions, as based on your attendance and the quality of your in-class contributions. 

Student Responsibility 

I will do everything in my power to help you learn, but it is up to you to make the most of this experience by coming to class prepared and ready to engage in discussion.  The following actions are all critical for your success in this class.  If there are barriers to you doing any of the following, please let me know.

ASK QUESTIONS & THINK CRITICALLY.  Throughout the course you should be asking yourself “Why are we doing this? Why does this method work? How is this related to other topics I’ve learned?” Ask questions in class, after class, during office hours, whenever! If you are confused or having problems with a certain section of the material talk to me about it.  It is best to clear up confusions right away, and if you have a question, almost certainly other members of class do to.  Be the brave one and speak up!  The whole class will benefit from your questions. 

DO THE ASSIGNMENTS. As you know, mathematics is not a spectator sport. At the same time, it can be a frustrating endeavor to engage in alone.  Think of it as a team sport! I strongly encourage, and may even require, you to work with your fellow students on homework assignments. 

SPEND TIME STUDYING. Willamette’s Credit Hour Policy holds that for every hour of class time there is an expectation of 3 hours of work outside of class. You should anticipate spending approximately 9 hours outside of class engaged in course-related activities.  To master the content of this course you will need to invest time and effort. Set aside time for both homework and studying. A good approach would be to spend approximately 2-3 hours on reading the sections of your text and making study notes,  and 3-6 hours doing the assigned problems each week.  This course is completely manageable, and you can do well in it as long as you engage from the beginning. 

How to Get Unstuck

Abstract Algebra has some of the most beautiful - and complex - definitions and results you will run into in your undergraduate mathematics career. It is natural to feel a little lost or confused at times. What should you do?  Discuss!  Ask questions! You can start conversations on the course Discord channel, bring questions up in class, arrange to meet with me, arrange to meet with other students, attend office hours, etc.  We now have the capability to meet in person or virtually, so there are even more ways to talk things through with me or your classmates.

Asking Questions in Class

Sometimes it's hard to know what exactly to ask, so here are some some ideas if you need inspiration:

• I'm a little confused on that last point, could you go over that again?

• How does this tie-into the overarching subject of this chapter?

• How did you know to take that step?  I don't think I would have known to do that.

• Can we quickly review the definition of <insert word here>?  I've forgotten what that means.

• Is it possible to quickly review why <thing 1> implies <thing 2>?  It's not obvious to me.

• How would I recognize what kind of <proof technique, calculation approach, etc.> to use here?  Are there particular clues to look out for?

Course Policies

Inclusivity Statement: Willamette University values diversity and inclusion; we are committed to a climate of mutual respect and full participation. My goal is to create a learning environment that is effective, equitable, inclusive, and welcoming. If there are aspects of the instruction or design of this course that result in barriers to your participation, assessment of your work, or your achievement, please notify me as soon as possible. I will gladly honor your request to address you by your affirmed name or gender pronoun. Please advise me of this at any point in the semester so that I may make appropriate changes to my records. If I accidentally use an incorrect gender pronoun or mispronounce your name when addressing you, please let me know, in whatever manner makes you comfortable, so that I can make every effort to correct that error. 

Students with disabilities are encouraged to contact the Accessible Education Services office in Smullin 155 at 503-370-6737 or Accessible-info@willamette.edu to discuss a range of options to removing barriers, including accommodations. 

Missed Classes:  Please make every effort to attend all classes and be an active participant in class activities and discussions. Much of your learning and understanding of new ideas will come from thinking and talking about them in class.   If for some reason you are unable to attend class or turn in an assignment, please let me know as soon as possible, preferably before the missed class or assignment. I know how important athletic commitments, personal events, and religious traditions can be, and will do my best to accommodate your personal needs.

Cell Phone/Screen Policy: Unless specifically stated for class use, no phones, laptops, iPads, or other devices which take your eyes off your fellow classmates and the class discussion are allowed. Electronic devices must be turned off or silenced during class meetings. If your phone goes off, or it is clear that you are using one of these devices during class, you will be responsible for bringing treats for the entire class at the next class meeting. 

Health: As a student, you may experience a range of challenges that can interfere with learning, such as strained relationships, increased anxiety, substance use, feeling down, difficulty concentrating and/or lack of motivation. These mental health concerns or stressful events may diminish your academic performance and reduce your ability to participate in daily activities. Willamette services are available and treatment does work. If you think you need help, please contact Bishop Health as soon as possible at http://willamette.edu/offices/counseling/. Crisis counseling is available 24/7 at WUTalk: 503-375-5353 and Campus Safety is available at 503-370-6911. Emergency resources are also available from the Psychiatric Crisis Center at 503-585-4949 and the National Suicide Prevention Lifeline at 1-800-273-8255. 

Willamette is a community committed to fostering safe, productive learning environments, and we value ethical sexual behaviors and standards. Title IX and our school policy prohibit discrimination on the basis of sex, which regards sexual misconduct including discrimination, harassment, domestic and dating violence, sexual assault, and stalking. We understand that sexual violence can undermine students success, and we encourage affected students to talk to someone about their experiences and get the support they need. Please be aware that as a mandatory reporter I am required to report any instances you disclose to Willamette’s Title IX Coordinator. If you would rather share information with a confidential employee who does not have this responsibility, please contact our confidential advocate at confidential-advocate@willamette.edu. Confidential support also can 4 be found with SARAs and at the GRAC (503-851-4245); and at WUTalk - a 24-hour telephone crisis counseling support line (503-375-5353). If you are in immediate danger, please call campus safety at 503-370-6911. 

Academic Integrity: Students of Willamette University are members of a community that values excellence and integrity in every aspect of life. As such, we expect all community members to live up to the highest standards of personal, ethical, and moral conduct. Students are expected not to engage in any type of academic or intellectual dishonesty and are encouraged to display integrity, trust, fairness, respect, and responsibility in all they do. Plagiarism and cheating are especially damaging and offensive to the College community as a whole. These acts involve intellectual dishonesty, deception, misrepresentation, and fraud, which inhibit the honest creation and exchange of ideas. Ignorance of what constitutes plagiarism shall not be considered a valid defense. Plagiarism and cheating may be grounds for failure in the course and/or dismissal from the College.   For more information, see http://willamette.edu/cla/catalog/policies/plagiarism-cheating.php. If you are unsure of what constitutes cheating, please ask me. These are the practices I expect you to follow in each of the components of the course: 

on the readings and homework assignments: You may, and are encouraged to discuss things with fellow students, and get help from your professor, your notes, or similar resources. However, your submitted written work should be your own or a collaboration with your assignment group. Copy/pasting sections of another's assignment,  reading and paraphrasing another source without proper citation, or providing your assignment to be copied by others, are violations of university policy. When working on a group assignment, all members of the group should contribute to producing all components of the finished product. Writing your name on work you did not contribute to in a substantive manner is a violation of university policy. 

on the use of Artificial Intelligence:  The ease of use and availability of artificial intelligence is increasing at a rate that is hard to fathom.  Open dialogue is crucial as we navigate this momentous shift in technology.  For this reason, consult your professor before using AI on any course assignment, and clearly cite AI as a source whenever you do use it.  Passing off the analysis, synthesis, or creative work of others as your own, whether surmised by an AI bot or not, is plagiarism.  Reasons you might consider using AI in an inappropriate way and why you should instead speak with your professor:

feeling crunched for time and under pressure: We have all felt overwhelmed at times and unable to complete all the tasks before us.  For that reason, there is a 24 hour grace period built in to almost all assignments and I am often able to offer even greater flexibility in due dates if you reach out to me within 24 hours of the assignment due date.  

do not see the benefits or purpose of the assignment:  It is never my intent to waste your time or make you do menial tasks that could easily be done by an AI bot.  If you do not see the educational value in an assignment, please ask me what the purpose is.  It is possible that I need to restructure the assignment given the technological tools available, or better explain the intended student learning outcomes.  

struggling to understand the material and synthesize new knowledge into an original product: When faced with an intellectual challenge, it is tempting to seek answers from the most readily available and anonymous source.  However, it is only by engaging with challenges that we learn.  If you need help processing a concept or solving a problem, turn to your classmates or me.  You will gain a deeper understanding, a greater connection with your peers, and an improved ability to advocate for yourself by collaborating with others in the Willamette community as opposed to turning to an AI bot for an instantaneous answer.  When you query an AI bot, you can not control the level of synthesis and analysis in the response.  Even if your intent was merely to summarize a text, review a concept, or perform some other low level task, the response you get may limit your ability to critically engage with the topic and shape your own opinions.  For this reason, I encourage you to engage with AI in a cautious and thoughtful way.  Given the impressive power of current AI bots and the speed at which they are improving, it is critical that you develop and strengthen skills that can not be easily supplanted by AI.  If you are using AI to avoid intellectual challenges now, what will you offer an organization or employer that can't be assigned to an AI bot in the near future?  

Intellectual Property & Privacy: Class materials and discussions including recorded lectures are for the sole purpose of educating the students enrolled in the course. The release of such information (including but not limited to directly sharing, screen capturing, or recording content) is strictly prohibited, unless the instructor states otherwise. Doing so without the permission of the instructor will be considered an Honor Code violation and may also be a violation of other state and federal laws, such as the Copyright Act. 5

DACA/Undocumented Student Advocate:  Willamette is committed to supporting our DACA/Undocumented students in a variety of ways. This year, Olivia Muñoz is the contact person for all DACA/undocumented students can provide those students with a number of external and internal resources that are available. Her contact information is email:omunoz@willamette.edu, Office: 3rd Floor UC, Phone: 503-370-6447.

SOAR Center Offerings (Food, Clothing, and School Materials): The Students Organizing for Access to Resources (SOAR) Center strives to create equitable access to food, professional clothing, commencement regalia, and scholarly resources for WU and Willamette Academy students. The SOAR Center is located in the Putnam University Center’s third floor, across from the Harrison Conference Room. The space houses the Bearcat Pantry, Clothing Share, and First-Generation Book Drive and is maintained by committed students and advisers. Please check www.willamette.edu/go/soar for current hours of operation and email soar-center@willamette.edu for any questions or concerns. 

Land Acknowledgement: We are gathered on the historic land of the Kalapuya, who today are represented by the Confederated Tribes of the Grand Ronde and the Confederated Tribes of Siletz Indians, whose relationship with this land continues to this day. We offer gratitude for this place, for those who have stewarded it for generations, and for the opportunity to study, learn, work, and be in community on this land. We acknowledge that our University's history, like many others, is fundamentally tied to the first colonial developments in the Willamette Valley. Finally, we respectfully acknowledge and honor past, present, and future Indigenous students of Willamette.