Math 344: Abstract Algebra
Fall 2004, MWF 12:10 – 1:00, R 11:00 – 11:50, MC 415
Instructor: Rich Maresh, Associate Professor, Mathematics Department Chair
Contact Information: MC 522, (796)-3655, (Home: 526-4988)
Office Hours: MWF 10–11, R 9-10, 2-3
Final Exam: Wednesday, 15 Dec 2004, 9:50 – 11:50 a.m.
Catalog Description: Study of selected algebraic topics such as: groups, rings, and fields; ring of integers, polynomials; field of real numbers, complex numbers; finite fields. Generally offered every other year. Prerequisite: grade of C or higher in 260.
Writing Course: MATH 344 is the “Writing” course for the mathematics program. We actually have students work on their proof writing skills in many mathematics courses, but this is the one formally designated to that end. You will note that in the “Student Outcomes” listed below, communication is emphasized.
Text: Elements of Modern Algebra (Fifth ed.), by Jimmie Gilbert and Linda Gilbert, Brooks/Cole Publishing, 2000.
Students shall investigate mathematical structures and properties that unify a variety of mathematical systems.
Students shall see how properties are derived logically from defining characteristics of a mathematical structure.
Students shall see the beauty, elegance, and usefulness of abstract mathematical structures.
Mathematics Program Course-Level Student Outcomes:
As part of an attempt to assess our mathematics program, basically to see that we are actually accomplishing what we want to accomplish, we have written a set of student outcomes for various courses taken by math majors. I will be keeping track of how well you manage to achieve these desired outcomes.
Students will review the basics of mathematical logic and proof, and the topics of binary operations and congruence classes, and will demonstrate an understanding of the properties of groups, rings and fields, and finally, of the applications of these structural concepts to the real and complex number systems and polynomials.
• Students will demonstrate an understanding of axiomatic-deductive systems. (Outcome 2.1)
• Students will demonstrate an ability to understand proofs. (Outcome 2.2)
• Students will be able to reason inductively. (Outcome 2.3)
• Students will be able to reason deductively. (Outcome 2.4)
• Students will demonstrate an ability to solve a variety of problems. (Outcome 3.1)
• Students will demonstrate an ability to solve problems in a variety of ways. (Outcome 3.2)
• Students will correctly use the language and notation of mathematics in their work. (Outcome 5.1)
• Students will demonstrate an ability to make oral presentations of their work. (Outcome 5.2)
• Students will demonstrate an ability to communicate mathematical content and proof in correct written form. (Outcome 5.3)
Course Philosophy and Procedures:
“Abstract Algebra” is a course about structure. The preface in your text says, “Such a course is often used to bridge the gap from manipulative to theoretical mathematics and to help prepare secondary mathematics teachers for their careers.” I think this is an accurate description of what this course is all about.
Human intellectual endeavor is mostly a matter of striving to find order and to communicate generalizations to others. Much of what we do in this course is in fact to try to find the properties that various mathematical “systems” have in common. We will see that what we call a “group” can consist of numbers under the operation of addition, complex numbers and multiplication, matrices and matrix multiplication, or functions and composition. We want to achieve a deep understanding of the structure that exists when mathematical objects (numbers, matrices, vectors, functions, etc.) have operations defined on them. In one sense, this course will be the most theoretical you have ever taken, and in another sense you will see that we are constantly looking at every day mathematical “things”.
My educational PHILOSOPHY in this course is that learning mathematics should be thought of as a process of adapting and organizing your quantitative world, not just discovering preexisting ideas imposed by others – and active process rather than a passive process. You should be developing mathematical structures that are more complex, abstract, and powerful than the ones your currently possess so that you be able to solve a wide variety of meaningful problems. You should also become more autonomous and self-motivated in your mathematical activities. You won’t be “getting” mathematics from me, but from your own explorations, thinking, and participating in discussions. You should come to see mathematics as an open-ended creative activity rather than a rigid collection of recipes. You should also develop a sense of skepticism, looking for evidence, example and counterexample, and proof, not just because a school exercise calls for it, but because of an internalized desire to know and understand.
It is extremely important in this course that you DO the HOMEWORK problems! The purpose of this course is both to learn content and to further develop your ability to solve problems and WRITE PROOFS. In order to do this you MUST do the work. There is an old adage that says a college student should expect to work at least 2 hours outside of class for each hour in class – and in a course like this you should at least expect this much work. It simply can’t be done with anything less, unless you are a genius of the first water!
Because part of the work of mathematicians is explaining their work to others, especially if they go into teaching, we will do quite a bit of presenting problem solutions orally, in class.
In addition to the routine homework assignments, I will be giving you on each Monday a PROBLEM SET, which will be due on the following Monday. While I encourage you to work with your colleagues on homework exercises, these problem sets should represent your own work. You may discuss the problems on the problem sets with ME but not with your classmates or other tutors. I will be asking you to sign a statement that each of these assignments you turn in is your own work. Especially because this is an “upper division” course, it is important that you demonstrate an independence as well as quality of work.
EXAMS pose a special problem for a course like this because the types of problems we will be working on are often not the sort that can be solved and written up in 10 minutes. So much of your grade will be gleaned from the quality of your work in the written problem sets to be turned in.
I generally use a GRADING scale of 90% for and A, 80% for a B, 70% for a C, 60% for a D.
AMERICANS WITH DISABILITY ACT:If you are a person with a disability and require any auxiliary aids, servicesor other accommodations for this class, please see me or Wayne Wojciechowski (MC 320, 796-3085) within ten days to discuss your accommodation needs.
In lower division courses, I typically construct a schedule that lists what we will be doing every day throughout the semester. In a course like this one, however, I find that I have to be more flexible than that, that I have to move through the material at a pace you can handle. We have one advantage here: this course is not a prerequisite for a subsequent course, so if we don’t get quite as far as I hoped, it is not a major issue, and I would rather have you learn well what we actually do, than to zip through material so we can say we “did it” even though you didn’t understand it.
That said, however, I still have content in mind, as indicated in the rough outline below. How far we get through this list has to do with how much time we have to spend on the early review material. I certainly hope to get through chapter 6, but the last two chapters are applications of the new content and would make a nice way to wrap up the course. We shall see…
Math 260 Review: (3-4 weeks)
[1.4] Binary Operations
[2.1] Postulates for the set of Integers
[2.2] Mathematical Induction
[2.4] Prime Factors, GCD
[2.5] Congruence of Integers
[2.6] Congruence Classes
Groups: (3-4 weeks)
[3.1] Definition of a Group
[3.3] Cyclic Groups
[4.1] Finite Permutation Groups
[4.2] Cayley’s Theorem
[4.3] Permutation Groups in Science and Art
[4.4] Normal Subgroups
[4.5] Quotient Groups
Rings, Integral Domains, and Fields: (2 weeks)
[6.1] Ideals and Quotient Rings
[6.2] Ring Homomorphisms
[6.3] The Characteristics of a Ring
Real and Complex Numbers: (2 weeks)
[7.1] The Field of Real Numbers
[7.2] Complex Numbers and Quaternions
[7.3] DeMoivre’s Theorem
Polynomials: (2 weeks)
[8.1] Polynomials over a Ring
[8.2] Divisibility and the Greatest Common Divisor
[8.4] Zeros of a Polynomial
[8.5] Algebraic Extension of a Field