THE COURSE:

The Abstract Algebra course (Math 155) is of monumental relevance to future teachers; Math 155 is an introduction to a part of mathematics that grew out of classical algebra (the study of polynomial equations) in the nineteenth century. Mathematicians of the twentieth century used the power of abstract algebra, to solve old problems of classical algebra and geometry, using new abstract techniques. Abstract algebra has evolved into a very rich field of mathematics research and applications; for example, some topics of abstract algebra are now applied in computer security.

During the semester the teaching and learning process will take the form of: lectures from the instructor; collaborative activities, primarily in the form of worksheets, where students work on specified problems, in the classroom; student oral presentations on topics to be assigned.

TEXT:

Abstract Algebra a Geometric Approach, by Theodore Shifrin, Prentice Hall

MATERIAL COVERED:

Chapter 1 - The Integers

1. Integers, Mathematical Induction, and the Binomial Theorem
2. The Euclidean Algorithm, Prime Numbers, and Factorization
3. Modular Arithmetic and Solving Congruences (Skip the Chinese Remainder Theorem)
4. Z_m, Rings, Integral Domains, and Fields

Chapter 3 - Polynomials

1. The Euclidean Algorithm
2. Roots of Polynomials
3. Polynomials with Integer Coefficients

Chapter 4 - Homomorphisms and Quotient Rings

1. Ring Homomorphisms and Ideals
2. Isomorphisms and the Fundamental Homomorphism Theorem

Chapter 6 - Groups

1. The Basic Definitions
2. Group Homomorphisms and Isomorphisms
3. Cosets, Normal Subgroups, and Quotient Groups

After we cover these topics we will proceed to show that there is a quintic that does not have solution by radicals.

GRADING:

HOMEWORK:

You are expected to work on the problems from the Textbook. I will indicate which ones need to be turned in for a grade. Other problems can be found here, they are to be turned in for a grade (visit this site regularly). Each problem will receive a grade from 0 to 12. I will allow redo's on problems receiving a grade of 6 or lower (except for level C problems), the maximum grade possible on a redo is 9. Homework is an individual activity, identical solutions will automatically receive a 0, with no possibility of redo's. Each problem must begin on a new page and must be clearly labeled; for each problem solution, I expect a clear line of reasoning that includes complete justifications and explanations, in which you use English sentences and mathematics, failure to do so will result in a low grade; a sole answer without a line of reasoning will automatically receive a 0, and you will not be allowed to redo the problem. Homework must be turned in on time, and no make-up's are allowed.

---

Challenging problems can be found here.

MINITESTS:

These are short one-hour tests. No makeup's are allowed unless you have a justified absence: away on a Drake sponsored trip (athletics, choir, band, etc.), ill (which must be justified with a doctor's note), or called home on an emergency. I will use attendance to decide borderline cases; a poor attendance record will result in a lower grade. If you do have a justified absence, please contact your instructor in advance.

---

A review sheet for the first minitest can be found here.

A review sheet for the second minitest can be found here.

TAKE-HOME EXAMS:

Each problem will receive a grade from 0 to 12. Exams are an individual activity, identical solutions will automatically receive a 0. Each problem must begin on a new page and must be clearly labeled; for each problem solution, I expect a clear line of reasoning that includes complete justifications and explanations, in which you use English sentences and mathematics, failure to do so will result in a low grade; a sole answer without a line of reasoning will automatically receive a 0, and you will not be allowed to redo the problem. Tests must be turned in on time, and no make-up's are allowed.

WORKSHEETS:

Below are the worksheets that were done in class with groups. Worksheets will not be graded, but you will be tested on the material that appears on these; you may also need to use some of the results in the worksheets for the take-home exams, or other homework. These worksheets are in pdf format and require Acrobat Reader to view them. .

Congruences Worksheet 1

Z_m Worksheet 2

Z_m Worksheet 3

Splitting Fields Worksheet 4

S₄ Worksheet 5

_{Last Modified: Tuesday, April 24, 2001}