MCS 314: Modern Algebra II (Spring 2013)

Course Description


Overview

Abstract algebra has its roots in the centuries-old efforts of mathematicians to find solutions to polynomial equations. Over the last few hundred years, as the subject has evolved, it has become the source of some of the most widely used tools in mathematics. Groups, rings, fields, and related algebraic structures are used in almost all areas of mathematics - analysis, topology, number theory, discrete mathematics - as well as in physics, chemistry, computer science, and other disciplines.

This course will continue where MCS-313 left off, focusing on rings, fields, and a branch of algebra called Galois theory, which can be used to settle such classical questions as the squaring of the circle, the trisection of an arbitrary angle by straight edge and compass, and the solution of polynomial equations of degree five or greater.

Course Objectives

  • to continue to develop the ability to think abstractly, make conjectures, and understand and write rigorous mathematical proofs
  • to learn and practice how to present mathematics orally
  • to continue to understand the basic philosophy, purpose and history behind the development of abstract algebraic structures
  • to understand Galois theory, which involves an an interplay between the most common algebraic structures - groups, rings and fields - to answer apparently unrelated classical questions
  • in particular, to understand how questions in one area (field theory) can be answered by translating those questions in another area (group theory) using correspondences called functors

Prerequisite

The formal prerequisite is MCS-313.

Class Web Site

All course materials will be available through the class web pages. The main URL for this course is http://homepages.gac.edu/~karl/courses/mcs314/13s/. I will give hard-copy hand-outs only to those students who ask for them.

Text

Contemporary Abstract Algebra, seventh edition, by Joseph Gallian. For the second half of the course, I will Probably distribute as set of notes on Galois theory which wil supplement Gallian. For each class session, I encourage you to read the pertinent portion of the text at least once beforehand and at least twice afterwards. Study the book with a pencil in hand. Make notes in it. Mark where you have questions. Do NOT try the exercises without reading the text; simply skimming the examples is not sufficient. You will find that it will be necessary to read the text several times before attempting any exercises.

Classes

Classes will be used for lectures, problem solving, discussions, and other fun activities. You should prepare for classes by doing the reading beforehand (reading assignments are posted on the Web), thinking about the problems in the text, and formulating questions of your own. You should also participate as much as possible in class. Class meetings are not intended to be a complete encapsulation of the course material. You will be responsible for learning some of the material on your own.

Should you need to miss a class for any reason, you are still responsible for the material covered in that class. This means that you will need to make sure that you understand the reading for that day, that you should ask a friend for the notes from that day, and make sure that you understand what was covered. If there is an assignment due that day, you should be sure to have a friend hand it in or put it in my departmental mailbox (in Olin 312). You do not need to tell me why you missed a class unless there is a compelling reason for me to know.

Two comments:

  • Attendance, both physical and mental, is required. I reserve the right to lower your grade if you habitually miss class or fail to participate in class.
  • Texting is not permitted in class. If you receive an urgent phone call, please just leave the class quietly and deal with the call.

Guidelines for writing proofs

Since one of the main objectives of this course is for you to understand how to write rigorous mathematical proofs, I am including the following link giving guidelines for writing proofs. I will use these guidelines when I am grading your mastery and exam problems (described below) as well as the proofs you write on exams.

Homework

I will assign homework at the beginning of each chapter by posting them on the web. The problems will be designated as practice problems, mastery problems, and exam problems:

  • Practice problems are problems to give you practice understanding and presenting mathematics. You should do these problems as you read the material and prepare for class. You will often be asked to present your solutions in class on the day we cover the material. You may work together on these problems, and they will be fair game for the tests.

  • Mastery problems are homework problems that are graded on an mastered/not mastered basis. You may turn in any individual homework problem in class whenever you think you have it solved, and I will return it to you by the next class, if not earlier (in which case I will put it outside my office in a folder I will supply for you), but normally with only an indication of whether it is acceptable or needs more work. The reason why I won't write much on the work I turn back to you is because I would like to talk with you face-to-face. If a problem needs more work, you should treat that as an invitation to come talk with me about it. Once you've done the additional work, you may turn the problem in again, attached to the original (you need to re-write it, but I would like to see what you did before). If your write-up is messy or grammatically incorrect, I will return it as not mastered.

    I only allow you to turn it on class days, since I want to force you to be careful about what you hand in, and not rely on me to make minor edits. I will give a deadline for each problem and until that deadline you may turn it in as often as there are class days.

    Mastery problems must be done individually, with no help from other people, books, or on the internet.

  • Exam problems are problems that each of you must do individually. You can think of these as miniature take-home tests; you are on your honor not to cheat by consulting other people, books, or on the internet. These problems will be graded using a 10 point scheme for each problem, and, except in extreme circumstances, you only have one chance to do each one. Exam problems are due on the day listed on the assignment.

Presentations

There will be two forms of presentations in this course:

  • Presentations of your solutions to the practice problems described above. I expect these presentations to be at most 5 minutes and will be graded on a 3-point scale.

  • A longer presentation of a topic from Galois theory during the second half of the semester. Your presentation will be anywhere from 20-50 minutes in length and will be over a topic we will agree upon and for which you will have some weeks to prepare.

Tests

There will be two exams: a mid-term and a final. These tests will be closed-book and closed-notes, though I will allow you a note-sheet or note-card of size to be specified later.

Course grade

The course components will contribute to your grade in the following proportion:

Practice problems/Class participation 10%
Mastery problems 10%
Exam problems 10%
Longer presentation 10%
Mid-term 30%
Final 30%

Academic Integrity

You are expected to to adhere to the highest standards of academic honesty, to uphold the Gustavus Honor Code and to abide by the Academic Honesty Policy. A copy of the honor code can be found in the Academic Bulletin and a copy of the academic honesty policy can be found in the Academic Polices section of the Gustavus Guide.

On practice problems, I encourage you to discuss problems and their solutions with each other. However, each of you should first make a real effort to solve each problem by yourself.

On mastery problems, exam problems, and the tests, you are expected to work completely by yourself, and to sign the honor pledge on each of these assignments. The first violation of this policy will result in a 0 on that assignment and notification of the Dean of Faculty. Further violations will result in failing the course.

Disability Services

Gustavus Adolphus College is committed to ensuring the full participation of all students in its programs. If you have a documented disability (or you think you may have a disability of any nature) and, as a result, need reasonable academic accommodation to participate in class, take tests or benefit from the College's services, then you should speak with the Disability Services Coordinator, for a confidential discussion of your needs and appropriate plans. Course requirements cannot be waived, but reasonable accommodations may be provided based on disability documentation and course outcomes. Accommodations cannot be made retroactively; therefore, to maximize your academic success at Gustavus, please contact Disability Services as early as possible. Disability Services is located in the Advising and Counseling Center.

Disability Services Coordinator Laurie Bickett (6286) can provide further information.

Help for Students Whose First Language is not English

Support for English Language Learners (ELL) and Multilingual students is available via the College's ELL Support staff person, Andrew Grace (7395). He can meet individually with students to consult about academic tasks and to help students seek other means of support. The ELL Support person can also consult with faculty members who have ELL and multilingual students enrolled in their classes. The College's ELL staff person can provide students with a letter to a professor that explains and supports academic accommodations (e.g. additional time on tests, additional revisions for papers). Professors make decisions based on those recommendations at their own discretion. In addition, ELL and multilingual students can seek help from peer tutors in the Writing Center.