Math 31 - Fall 2011

Dartmouth College

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"Many who have never had occasion to learn what mathematics is confuse it with arithmetic, and consider it a dry and arid science. In reality, however, it is the science which demands the utmost imagination [which is more than just making things up] ..... It seems to me that the poet must see what others do not see, must look deeper than others look. And the mathematician must do the same thing." -- Sonya Kovalevsky.

Welcome to Math 31, Topics in Algebra!

 Course Goals Course Structure Honor Principle Other Issues Grading

Course Goals

Through this course, you will develop (greater) comfort and facility with:

• Learning mathematics from sources other than your instructor (such as your textbook, your peers, and your homework assignments)
• Communicating mathematics, with words and symbols, in a variety of contexts
• Writing mathematical proofs that are correct, clear, and concise
• Mathematical abstraction -- both abstract constructions in themselves, and the relationship between the abstract constructions and concrete examples.
• In order to develop and demonstrate your ability in the above skill areas, you will have to learn some new mathematics in Math 31, as well. Math 31's content focuses on the algebraic structures known as groups and rings, which show up in almost all areas of math, as well as physics, chemistry, biology, and even art! So, this course will also focus on answering the following questions:

• What does it mean for two groups (or rings) to be "the same?" How can we tell if two groups are the same?
• How can we make new groups? How do we know when we have found all possible groups?
• How can we make groups into rings, or rings into fields? What properties get preserved? What ones get lost?
• How does the material in Math 31 (groups and rings) connect with your other courses, past and future?
• Please note that none of the above goals is "Preparing you for Math 81." If you are planning to take Math 81 in the future, I strongly advise you to take Math 71 instead of Math 31. I am happy to meet with you individually to discuss whether Math 71 or Math 31 is more appropriate for you.

Course Structure

Each of the components of this course is designed to help meet one of the above goals. If you find that some aspect of the course structure is not helping you to achieve the goals (or answer the questions) above, please tell me! I would be more than happy to work with you so that we can figure out a way for you to achieve the course goals.

This course will include: Reading Assignments, Class Meetings, Quizzes, Written Homework, Midterm Exam, Presentation, Final Exam.

We will spend time in class discussing how to read a math textbook; it's very different than reading for a political science class, for example!

The Reading Assignment serves (at least) 4 purposes:

• You will develop your ability to learn math from the textbook, not just from lecture.
• Math makes much more sense the second time around, so reading the textbook the night before class will prepare you to learn well in class the next day.
• Your comments and questions will help me focus class time on points you found confusing.
• Class time will be spent primarily on activities that complement and enhance the textbook, rather than imitating the textbook.

• Class Meetings
Math 31 will meet in Kemeny 006, at 12:30 PM on Mondays, Wednesdays, and Fridays. We will meet for our X-hour on Tuesdays at 1:00 PM. We will use all X-hours, and I will pass around a sign-in sheet at the beginning of each class. Because we will often spend class time on material that is not in the textbook, or that the textbook presents differently, attending class will be very important to your success in this class. We will also frequently spend class time on collaborative activities, so your attendance will also be important to your classmates' success.
Therefore, failure to attend class regularly, either mentally or physically, will adversely affect your grade.

Out of courtesy to your fellow students, and to me, please focus your attention on Math 31 during class time! In particular, please refrain from cell phone use (including texting), computer use (unless you take notes on your computer), working on homework for another class, or talking about Friday night's awesome party. Our classroom is not the time or place for these activities, fun as they are.

Class time on MWF will generally be spent clarifying questions from the reading, looking at the topic from a different perspective, and working through examples. Most classes will include some lecture, but they will also include activities for YOU, either individually or in groups. This way you will have a chance to try your hand at using the definitions and theorems to solve problems while I am around to answer questions, and before you have to do so on the homework or exams.

For the first half of the course, the X-hours will be used to develop your facility and comfort with some crucial topics that are not tied to any particular section in the book, such as proof-writing and the various notations and symbols used with groups and rings. During the second half of the course, the X-hours will be devoted to your presentations (more on these below!)

Quizzes
Each week, we will have a brief (5-10 minute) "vocabulary" quiz at the beginning of class on Monday. The purpose of the quizzes is to encourage you to learn the important definitions and theorems thoroughly enough that you aren't constantly running to the textbook, or to your notes, to check them. I might ask you to state a definition or a theorem, give an example, or perform a simple computation. If you have been reading the textbook, doing the homework, and attending class, the quizzes should not be difficult.

If you will be unable to attend class, please make arrangements with me to take the quiz early. Quizzes may not be taken late; however, you may drop your lowest quiz score.

Written Homework
There will be weekly written homework assignments. These will generally be posted on Friday and collected at the beginning of class on Friday of the following week. No late homework will be accepted, but you may drop your lowest homework score.

You are encouraged to discuss the homework problems with your classmates, but you should write up the homework on your own. It is a violation of the honor code to copy on the homework. In other words, the written assignment you turn in must be in your own handwriting (or typing), in your own words, and reflect your own understanding of the material. To this end, I recommend that you take a few notes when discussing a homework problem with your classmates, and then write up the homework on your own, later, in order to remove the temptation of writing down your classmate's explanation rather than your own thoughts.

In my view, the written homework is primarily a vehicle for learning the course material, and for helping you prepare for exams. To that end, the homework each week will include a few problems designed to check your understanding of the topics we studied that week, but it will also include several problems requiring a deeper insight or a synthesis of several topics. Occasionally the homework may introduce a concept we haven't discussed in class. I encourage you to review your returned homework assignments regularly and often, and to meet with me if there are topics you still don't understand after finishing the homework!

Starred Homework Problems
Each week, (at most) one problem on the written assignment will be starred (e.g. Chapter 6, problem 4*). These problems will usually be more challenging than the rest of the homework, and they will require you to write a proof. The starred problem should be turned in on a separate page from the rest of the written homework. I will grade these problems myself, on a credit/no credit basis (but with comments so that you know where the problems are). If you receive a score of no credit, you can resubmit the problem the next week along with that week's homework assignment, and you can continue this process until you receive credit. The primary point of this is to make sure that everyone learns to write good proofs. The secondary point of this is to encourage you to solve problems which might require more than one week of thinking -- but not all of the starred problems will be that hard!

Presentations
In order to develop your ability to communicate mathematics verbally, you will be divided into groups of 3 or 4 (based on interests) and each group will be asked to learn about a supplemental topic and teach the rest of the class about their topic in a 20-minute presentation. The presentation topic should be related to the material we're studying in Math 31 -- some possible topics include applications of groups and rings to chemistry, art, number theory, cryptography, error-detecting codes, or physics. You could also give a presentation about some aspect of the history of abstract algebra -- the development of the concept of a group, the role Emmy Noether played in developing ring theory, or the problems that motivated the invention of certain rings such as the quaternions or the Hamiltonians, to give a few examples. Another possibility would be a mathematical topic that isn't on the syllabus, such as the difference between principal ideal domains and unique factorization domains, or the Holder program for finite groups, or a description of all the finite fields of characteristic p.

The topics listed above are definitely not the only possible presentation topics! If none of the topics listed above catches your attention, or if you have another idea for a presentation topic, please let me know. Any topic is fine, as long as the entire group is excited about it, and as long as it's related to Math 31 material.

My goal with this assignment is to ensure that everyone learns something from each presentation. I would rather not have you feel that the 7 presentations you didn't participate in were a waste of your time.

I expect each group to meet with me several times before giving their presentation. The dates given below are the latest possible dates for each meeting:

• Friday, Oct. 7 Choose a date for your presentation.
• Fourth Friday before presentation Confirm a topic for your presentation; meet with me to get resources.
• Two weeks before presentation Narrow the focus of your presentation; preliminary discussion of the format of the presentation and the homework problems you will choose.
• Friday before presentation Finalize choice of homework problems; final review of presentation content and format.
• The above schedule is a minimum. If you would like to meet more often, I would be more than happy to!

Exams
The midterm exam will include a take-home and an in-class portion. The in-class portion will occur during our normal class time on Friday, October 28. The take-home portion will be distributed the same day and will be due at the beginning of class on Wednesday, November 2.

The final exam will be administered during our final exam block: Saturday, December 3, 8:00 to 11:00 AM.

I will talk more about the exams in class as we get closer to the exam dates.

Honor Principle:

Dartmouth students are expected to adhere to the honor principle. In this course that means:

On Homework: Working together is permitted and encouraged, but no copying. In other words, the written assignment you turn in must be in your own handwriting (or typing), in your own words, and reflect your own understanding of the material. This means you cannot simply copy down the solution arrived at by the group, even if you were a member of the group. To this end, I recommend that you take a few notes when discussing a homework problem with your classmates, and then write up the homework on your own, later, in order to remove the temptation of writing down your classmate's explanation rather than your own thoughts.
If you do work in a group or receive help from a tutor or friend, state that on your homework, and include their names if they are in the class.

On Quizzes and Exams: No help will be given nor received from any person other than the instructor. Different exams may have different rules regarding the resources allowed. These rules will be printed on the exams themselves and stated in class before the exam, with enough time to prepare or study the appropriate resources.

If you have any questions as to whether some action would be acceptable under the Academic Honor Code, please speak to me, and I will be glad to help clarify things. This is a case in which it is definitely better to ask permission rather than forgiveness.

Disabilities and Religious Observances:

Students with disabilities enrolled in this course and who may need disability-related classroom accommodations are encouraged to make an appointment to see your instructor before the end of the second week of the term. All discussions will remain confidential, although the Student Accessibility Services office may be consulted to discuss appropriate implementation of any accommodation requested.

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 your instructor before the end of the second week of the term to discuss appropriate accommodations.