# general information

Title: MA4140: Algebraic Structures

Time: TR 8:00-9:15AM

Location: Hyde 316

Course Wiki: http://ma4140.wikidot.com

# instructor information

Instructor: Dr. Dana Ernst

Office: Hyde 312

Phone: 603.535.2857

Email: ude.htuomylp|tsnrecd#ude TOD htuomylp TA tsnrecd

Office hours: MWF at 11:00-12:00PM and T at 1:30-2:30PM (or by appointment)

Webpage: http://oz.plymouth.edu/~dcernst

# course information & policies

## prerequisites

MA3110: Logic, Proof, & Axiomatic System and MA3120: Elements of Linear Algebra (may be concurrent).

## catalog description

Designed to study the properties of groups, rings, ideals and fields, and develop selected topics. (That's sufficiently vague!)

## course content

This course is an introduction to abstract algebra. Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras [1]. For more information, see the Wikipedia article located here. We will spend most of our time studying groups, but we will have an opportunity to explore additional topics in your group projects. We will take an axiomatic approach (definition, theorem, and proof) to the subject, but along the way, you will develop intuition about the objects of abstract algebra, pick up more proof-writing skills, and skills that enable you to better read, understand, and communicate mathematics. We will also discuss how the field of abstract algebra fits into the broader "picture" of mathematics and take a look at some applications. The emphasis of this course is on your ability to *read*, *understand*, and *communicate* mathematics in the context of abstract algebra.

## text

We will be using the free open source textbook Abstract Algebra: Theory and Applications by Tom Judson. You can either download the PDF of the book here and be responsible for the printing the book yourself or you can buy a printed version of the book (published by the VCU Mathematics Department) from Amazon (go here). I will refer to the book available on Amazon as the "VCU version of the book."

*Warning:* You should be warned that there are two problems with the VCU version when compared to the PDF available on the web page. The first is that the PDF starts with chapter 0 and the VCU version starts with chapter 1. The content is the same, but all the chapters are off by 1. This shouldn't be a big deal once you are aware of it. The second thing is that the VCU version used a different font and as a result, the content on each page does not match up. That is, a given sentence appears in both books, but probably not on the same page. The PDF version of the book will be the canonical version. So, any reference I make to the book will be with regard to the PDF version.

I expect you to be *reading* the textbook. I will not be covering every detail of the textbook and the only way to achieve a sufficient understanding of the material is to be digesting the reading in a meaningful way. You should be seeking clarification about the material in the textbook whenever necessary by asking questions in class or posting questions to the course forum.

## writing connection (WRCO)

In order to communicate effectively, students need to learn the conventions of their own discipline or profession. They need to learn how to write like an educator, a social worker, a biologist, an historian, or a literary critic, for example. Students take a three-credit Writing course within their major that contains significant writing experiences appropriate to the discipline. These experiences should be based on Writing Across the Curriculum activities, for example, free-writing, outlining, writing multiple drafts, responding to feedback, and creating a finished product. In addition to extending the process of developing writing skills, Writing courses also emphasize writing to learn in the discipline.

MA4140 satisfies the WRCO requirement of the PSU General Education Program. One of the major goals of the course is for students to become competent and confident in reading and writing technical prose that occurs in the discipline of mathematics. The course develops methods of reasoning required to prove theorems and explain solutions to abstract mathematical problems. Students also gain proficiency in the language of abstract mathematical proofs. Writing proofs of theorems or other statements allows students the opportunity to practice logical thinking and document rigorous logical arguments. As students become increasingly skilled in thinking clearly and ordering their thoughts, they should gain greater aptitude in writing clearly and concisely. Students will complete daily homework assignments in which mathematical writing composes the majority of the work. Students are expected to use proper grammar and write in complete sentences.

## homework

The homework comes in two varieties:

### 1. textbook exercises

I will assign homework from the textbook each class session. Typically, an assignment will be due 1-2 classes after it is assigned. I will always make it clear when a given assignment is due; so there should never be any confusion. Homework assignments will be posted here. *Tip:* if you click on the `RSS` link on the homework assignments page, you can set up to be notified when I add a new assignment. Homework from the textbook will not usually be collected, although if necessary, there may be exceptions to this. Instead, you will be required to present solutions to these exercises on the board. See "class presentations" below for more information.

### 2. Sage labs

We will make use of Sage, a free open source mathematics software system licensed under the GPL. It combines the power of many existing open source packages into a common Python-based interface creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab. You will need to sign up for a free Sage Notebook account at http://www.sagenb.org/register (which you will then be able to use in all of your mathematics courses). There will be roughly 5-10 Sage labs that will be turned in via email. For most, if not all, of the Sage labs, you will be allowed (in fact, required!) to work in groups of 2-4. We will spend class time some time during the first couple weeks getting acquainted with Sage. This portion of your overall grade is 10%. The Sage labs will be posted on the Sage labs page.

## class presentations

Typically, I will lecture for the first 40-45 minutes of each class session. The remaining class time will be reserved for students to present solutions to the textbook exercises due that day. Though the atmosphere in this class should be informal and friendly, what we do in the class is serious business. In particular, the presentations made by students are to be taken very seriously since they spearhead the work of the class. Here are some of my expectations:

- In order to make the presentations go smoothly, the presenter needs to have written out the proof in detail and gone over the major ideas and transitions, so that he or she can make clear the path of the proof to others.
- The purpose of class presentations is not to prove to me that the presenter has done the problem. It is to make the ideas of the proof clear to the other students.
- Presenters are to write in complete sentences, using proper English and mathematical grammar.
- Presenters should explain their reasoning as they go along, not simply write everything down and then turn to explain.
- Fellow students are allowed to ask questions at any point and it is the responsibility of the person making the presentation to answer those questions to the best of his or her ability.
- Since the presentation is directed at the students, the presenter should frequently make eye-contact with the students in order to address questions when they arise and also be able to see how well the other students are following the presentation.

Since we have so many students enrolled in the course, we will form groups of two to be responsible for each other. Each pair will present as a team and receive the same grade. We will discuss how these teams are to be formed on the first day. If necessary, I will allow "divorces" and "remarriages." I will be grading presentations both on the content of their proof and on the quality of the presentation. I will probably grade the quality more harshly as the semester wears on, since I expect your presentation skills to develop with practice. However, your presentation grade will be most based on the frequency with which you present, as well as your level of interaction during other's presentations. Your presentation grade is worth 25% of your overall grade, which is largest percentage of any category.

I will always ask for volunteers to present proofs, but when no volunteers come forward, I will call on someone to present their proof. Each student is expected to be engaged in this process. The problems chosen for presentation will come from the assigned homework.

## exams

There will be two midterm exams and a cumulative final exam. All exams will consist of an in-class part and a take-home part. The in-class portion of the midterms will take place on Thursday, March 4 *Tuesday, March 9* and Thursday, April 15 *Tuesday, April 20*. The take-home portion of the midterms will be due by 5:00PM on *Friday, March 12* and *Friday, April 23*, respectively. The in-class portion of the final exam will take place on *Tuesday, May 18 at 8:00–10:30AM*, and the take-home portion of the final exam will be due by *5:00PM on Friday, May 21*. Each exam will be worth 15% of your overall grade. Make-up exams will only be given under extreme circumstances, as judged by me. In general, it will be best to communicate conflicts ahead of time.

## group projects

There will be one group project that will be due the last week of classes. Groups (consisting of 4-5 students) will be required to explore an additional topic and to report their findings. The project will have two components:

- written submission to the course wiki;
- oral presentation to the class.

Both portions of your project should contain three main components:

- description of an advanced topic
*directly*related to abstract algebra; - summary of a direct application of the topic to another area of mathematics;
- explanation of the relevance of the topic or application.

You will be provided with a list of topics to choose from. More information can be found on the group projects page. The group project is worth 10% of your overall grade.

## online participation

During the semester, you will use the wiki to:

- Ask questions of your Professor and fellow students and post responses to these questions.
- Collaboratively post content to chapter summaries consisting of definitions, theorems, and standard examples for use on the in-class portion of exams.
- Post group projects.

Part of your grade will be based on your participation in the online wiki. I will be able to see what contributions you have made to the site, and grade you accordingly. Each individual will be required to make at least two contributions to the chapter summaries prior to each exam. For each in-class exam, I will print out the current chapter summaries which you will be allowed to consult during the exam. So, the more content you post to the chapter summaries, the more information you will have available to you. Each entry in the chapter summary should be written in your own words, not "cut and pasted." Furthermore, you should use proper mathematical notation using $\LaTeX$. Your grade for this portion of the course is worth 10% of your overall grade and is "pass/fail." However, if you make substantial contributions to the chapter summaries or course forums, I will increase the weight of this portion to 15% and subtract 5% from the weight of your weakest category (i.e., exam grade, class presentation grade, Sage homework grade, etc.).

In addition, each group will be responsible for writing up their group project on the wiki. Again, I will be able to track your progress, see what contributions you have made to the site, and grade you accordingly.

## attendance

Regular attendance is expected and is vital to success in this course, but you will not be explicitly graded on attendance.

## basis for evaluation

Here is a summary of how you will be evaluated.

category | weight |
---|---|

presentations | 25% |

Sage labs | 10% |

midterm exam 1 | 15% |

midterm exam 2 | 15% |

online participation | 10% |

group project | 10% |

final exam | 15% |

# additional information

## Group Explorer

Occasionally, I will make use of Group Explorer, which is mathematical visualization software for the abstract algebra classroom. It helps the user visualize groups, build intuition about group theory, and enable experimentation with groups. Group Explorer can be downloaded for free from http://groupexplorer.sourceforge.net/.

## student handbook

The PSU Student Handbook addresses policies pertaining to students with disabilities, religious observation, honor code, general conduct, etc. The Handbook can be found here.

## ACT for Growth

All teacher education majors are subject to the Areas of Concern/Targets for Growth policy, which is located here.

# closing remarks

When does the learning happen? It might happen in class, but most likely it happens when you sit down to do your homework. Most of you can follow what I do on the board, but the question is, can you do it on your own? To learn best, you must struggle with mathematics on your own. It is supposed to be difficult. However, if you are struggling too much, then there are resources available for you. Work together and help each other learn. Use the course forums! I am always happy to help you. If my office hours don't work for you, then we can probably find another time to meet. It is your responsibility to be aware of how well you understand the material. Don't wait until it is too late if you need help. *Ask questions!*

Aside from the obvious goal of wanting you to learn a few things about abstract algebra, one of my principal ambitions is to make you the student independent of me. Nothing else that I teach you will be half so valuable or powerful as the ability to reach conclusions by reasoning logically from first principles and being able to justify those conclusions in clear, persuasive language (either oral or written). Furthermore, I want you to experience the unmistakable feeling that comes when one really understands something thoroughly. Much "classroom knowledge" is fairly superﬁcial, and students often find it hard to judge their own level of understanding. For many of us, the only way we know whether we are "getting it" comes from the grade we make on an exam. I want you to become less reliant on such externals. When you can distinguish between really knowing something and merely knowing about something, you will be on your way to becoming an independent learner. Lastly, it is my sincere hope that all of us (myself included) will improve our oral and written communications skills.