Math 281 Modern Algebra II

Meeting Times: Monday, Wednesday, and Friday, 12:00 - 12:50 pm
Location: Hirt 313
Office Hours: Monday 9-10 and 1-2, Tuesday 11:30-12:30, Wednesday 11-11:45, Thursday 8:30-9:15, Friday 9-11
Prerequisite: Math 280

This is the second semester of a year long sequence on the study of algebraic structures. Course topics include rings, fields, an introduction to Galois theory, symmetry, the Sylow theorems, and finite simple groups.

On successful completion of the course, students should be able to:

• provide the definitions of algebraic objects, and know some examples of each.
• develop abstract and critical reasoning by studying and writing mathematical proofs.
• understand the connection between modern algebra and other branches of mathematics.
• relate the material learned in this course to prerequisite courses.
• recognize algebraic structures and objects in everyday situations.
• learn about the historical development of modern algebra.

We will be using Contemporary Abstract Algebra, 8th Edition, by Joseph A. Gallian. An older edition of the text would be fine. No other texts or materials are required. You will not be required to bring the text to class, so an electronic version is acceptable. This is the same text we used for Modern Algebra I.

The homework assignments given in class will be similar to those from Modern Algebra I. You will have at least one week to complete each assignment. You are free to work together on your assignments, but everyone must submit their own work, in their own words. Late homework will be accepted with a 10% per day penalty, and will not be accepted after the graded assignments are returned.

We will have two midterm exams and a final exam. The final exam will be cumulative, while the midterm exams will focus on more recent material. Both exams will be based on homework problems and the suggested textbook problems that do not need to be turned in.

Exam Dates:
Midterm Exam I: Wednesday, February 28
Midterm Exam II: Wednesday, April 18
Final Exam: Thursday, May 10, 10:30 - 12:30

Your final grade will be calculated as follows:

• Midterm Exam Average: 30%
• Assignments: 30%
• Conference: 25%
• Final Exam: 15%

Your letter grade will be determined according to the department grading scale:

This is your chance to investigate a topic in algebra on your own, and share what you've found with the rest of the class. Some information is provided below, but more detail on expectations will be discussed during the Conference Workshop on Friday, March 2. We'll use this class to discuss possible topics, where to find references, and more.

There will be three major components of your conference grade:

• Abstract (10%)
  
  Before beginning work on your chosen topic, you'll be required to submit a brief abstract (one to two paragraphs) that describes exactly what you'll be researching and presenting. This need not be detailed, but should give the reader a clear idea of what to expect from your talk and submitted paper.
  
  The abstract for your talk will be due Friday, March 16.
  
  If the abstract is "accepted", you will receive full credit for the abstract and may begin work on your seminar talk and write up. If the abstract is not approved, you may revise and resubmit (with no grade penalty) until accepted. The goal will be to choose and describe a topic that is suitably challenging for this level of mathematics, but also possible to explain or demonstrate in a class talk.


• Conference Paper (60%)
  
  Your conference paper should be detailed narrative of your talk, including proper citations where appropriate. Some guidelines for the conference paper will be provided.
  
  A draft of your paper will be due Wednesday, April 11. Suggestions for revision will be returned.
  
  A final paper will be due Friday, April 27.
  
  The final version of the paper must be typed and include all references. Only the final version will be used for grading. Your papers will be collected and published in the Math 281 Conference Proceedings.


• Conference Talk (30%)
  
  During the last week of the semester, you'll be sharing your research with your colleagues. Each talk will be twenty minutes, including a few minutes for any questions. While not required, you should make use of the chalkboard, slides, handouts, props, videos, etc to help clarify your presentation.
  
  You will not be required to submit an outline or plans, but I would be happy to review your strategy or assist in constructing a strong talk.

Other notes:

• Your chosen topic is not expected to be original research. The paper and talk should be expository in nature, meaning you will simply need to choose a topic related to abstract algebra. Ideally, your topic will be something you are personally interested in, and which we have not already seen in class. I can assist you in selecting a suitable topic.
• Throughout your paper and talk, you may assume anything we've seen in class is prerequisite material. That is, you will not need to define words like group or field, and you will not need to provide a citation for known facts like Lagrange's Theorem.
• Citations in mathematics are used for efficiency. If your paper makes use of a lengthy proof or other result you found in an external source, you do not need to repeat the entire passage. Simply mention in your paper where the proof or result could be found. Examples will be provided in class.

You're free to come up with your own topic for your presentation, but if you're having trouble thinking of one, here's a list of journal articles and other sources you might find something in. Summarizing an article, along with any necessary background material, in a 20 minute talk is a challenge, so keep that limitation in mind when choosing a topic.

• College Mathematics Journal (MAA): You won't be able to access the articles unless you're a member of the MAA (there's a student discount!) but you can read through the abstracts. If there's an article you're interested in, just let me know and I'll get it for you.
• American Mathematics Monthly (MAA): MAA again, so if you find an interesting article, I'd be happy to get it for you.
• Mathematics Magazine (MAA): Just join the MAA already.
• Harvard Mathematics Review: There's only six issues, but each is filled with articles written by and for undergraduates.
• Rose Hulman Undergraduate Math Journal: Lots of issues to look through, though many of these will be in applied math areas (that's still a possibility for a conference talk, you'll just need to ensure it's related to algebra).
• Involve: A math journal that publishes papers with at least one undergraduate author. These may be a little higher level, but a good summary of an article would be a great talk.
• arXiv.org: Ok, you probably won't find your topic here, but have a look at some great (free) math!

In keeping with college policy, any student with a disability who needs academic accommodations must call Learning Differences Program secretary at 824-3017, to arrange a confidential appointment with the director of the Learning Differences Program during the first week of classes.

This course supports the mission of Mercyhurst University by creating students who are intellectually creative. Students will foster this creativity by: applying critical thinking and qualitative reasoning techniques to new disciplines; developing, analyzing, and synthesizing scientific ideas; and engaging in innovative problem solving strategies.