Abstract Algebra MAT 205, AUBG, Math & Sci Dept, Spring 2011

Abstract Algebra MAT 205, AUBG, Math & Sci Dept, Spring 2011


Abstract Algebra MAT 205, AUBG, Math & Sci Dept, Spring 2011

MAT 205: Abstract Algebra

required for Math Major & Minor;

Lecturer: Tatiana Gateva-Ivanova

office 317 NAB, phone: 491

Classes: Tuesday–Thursday 12:30-13:45, BAC 326

Seminar and Office hours: We 16–17:15 R326BAC

Textbook: Michael Artin, Algebra, Prentice Hall, 1991

Course description: The course offers an introduction to some basic algebraic structures-groups, rings, integral domains, and fields. As an application, the general approach is tested and exemplified by some important mathematical objects, as symmetric groups, some matrix groups and rings, ring of integers, polynomial rings. This is a course with an emphasis on learning to understand, construct and present proofs. One of the goals of the course is learn to think creatively, be able to attack a problem you have not seen before, develop tools for that, develop a mathematical model for a given “real life” situation.

At the end of the course you will extend your picture of the mathematical world, where various phenomena will be placed on their right position. Such an approach allows uniform and elegant examination and interpretation of various mathematical problems.

The course requires an accompanying weekly seminar.

Prerequisites: MAT 105 or equivalent.

Any other MAT course will be an advantage.

Expanded Description: (sections from the textbook we will cover)

ContentNo of Lectures

1. Introduction to Group Theory [Ch. 2] 20% of the course 6

The definition of a group, initial concepts and properties [2.1]

Examples of groups: Cyclic groups, Symmetric groups

Subgroups. The Klein four group. The quaternion group [2.2]

Homomorphisms and Isomorphisms., [2.3, 2.4]

Cosets and Lagrange’s theorem [2.6]

Modular arithmetic* [2.9]

Normal subgroups, Quotient groups;

First Isomorphism Theorem [2.10]

2. More Group theory [Ch. 5, Ch 6] 20% of the course 5

Computation in the Symmetric Group [6.6]

Symmetry and Groups

Action of a group on a set [5.5, 5.6]

Sylow theorems. Applications [6.4]

Classification of groups of small orders.

3. Rings and Fields [Ch. 10] 20% of the course 6

Rinds and Fields – initial concepts and properties [10.1]

Examples, Modular arithmetic

Homomorphisms and ideals [10.3]

Quotient rings.

Fiirst, second and third isomorphism theorems [10.4]

Integral domains and fraction fields* [10.6]

Maximal ideals [10.7]

4. Polynomials [Ch. 11, Ch. 13 & Ch. 14] 40% of the course 8

Polynomials in one variable

Factorization of integers and polynomials

Roots of a polynomial – Splitting field

Polynomials with integer and rational coefficients

Primitive polynomials – Gauss’s lemma, Eisenstein Criterion

Polynomials in several variables, Symmetric polynomials

Polynomials with real and complex coefficients

Finite fields *

Total25 lectures

Course objectives:

•To provide students with a theoretical framework, basic skills and problem solving abilities in the area of some fundamental algebraic structures.

• To facilitate students in mastering their analytical skills, logical critical thinking ability, and problem solving capability in the topics discussed.

• To develop students’ capability to read, write and present mathematics

Learning objectives: The students are expected to improve their ability to deal with variety of mathematical problems by detecting an adequate model, then applying appropriate collection of tools to solve it effectively, and at the end to present and discuss the final results.

Instructional Modes: Lectures, class discussions and weekly seminar sessions will be used to study and analyze the topics. Student involvement will be an ongoing affair – competent class activity will be strongly encouraged and awarded.

Assessment: Your grade will be formed by

final exam 40%of the final grade (maximum of 100 points)

two quizzes (midterms) each gives 25% of the final grade (each maximum of 100 points )

3 or more pop quizzes cumulative grade, 10 points maximum will be added to the total grade

Participation in class /or presentation on the seminar can add to the total grade maximum 10 points

competent performance in a class may give up to 1 points, or demonstration (short talk) on the board during a seminar may give up to 2 points.

Total = 40% F +25%M1+ 25%M2 + cumulative points for popquizzes (max 10pts)

+ Active Participation (max 10 points) Max total 110

Points/Grade Map:

A 100-110 A- 93-99 B+ 86-92 B 80-85 B- 75-79

C+ 67-74 C 60-66 C- 55-59 D+ 48-54 D 40-47.

Every class session I will ask the students to participate actively in the solutions of the problems. For each class there will be home assignment

(i) reading – text material from the book and (b) several examples from the book. The significant example will be discussed in class and on the weekly


• The popquizzes will be scheduled sporadically (typically at the end of the class) and will last 20-30 minutes.

Each popquiz will give in total 4-5 points. Some of the questions will require good knowledge and understanding of the definitions and important statements from the theory. The grade for the popquizzes is cumulative, based on the total number of points received for (all) popquizzes.

The maximal total number is 16, but only 10 points participate in the final grade.. There is no make up for popquizzes.

• Each quiz will hold 90 min and will consist of 5 or more problems. The solution of the problems require good understanding of the material, analytical skills, and creative thinking.

Each exam (quizzes and final) is designed for 130 points max, (30 are bonus) but only 100 points participate in the final grade.

• There will be a make up for each quiz on a date and time fixed by the professor.

• The final will be comprehensive, i.e., over all the material covered by the course.

Attendance and participation in class:

I expect you to come to class prepared (having read and the assigned text, knowing well the material from the previous lecture) and with written homework, I will strongly encourage active participation during the class.

Attendance will be checked regularly, starting from the first week of the semester.

An excuse for an absence should be asked from the professor before the class.

In case of absences due to medical reasons, the students are kindly requested to inform the professor via e-mail (before the class). It is a responsibility of the students to take care and prepare with the material covered during their absence.

Important dates:

• Midterm 1: February 18, 17:45-19:30, Auditorium, MainBuilding

• Make up M1: March 1 8, 17:45-19:30, AuditoriumMainBuilding

• Midterm 2: April 15, 17:45-19:30, Auditorium, MainBuilding

• Make up M2 April 26/29 (during the regular class the date will be fixed later)

• Popquizzes (4 or more ) will be scheduled sporadically and will last 20-30 minutes, at the end of the class . There is no make up for popquizzes.

Approximate dates for the first two are:

P1: Febuary 3; P2: February 14.

Academic Honesty Policy. Students are expected to adhere to the Academic Honesty Policy stated in the Catalog and the Student Handbook. Consequences for violations are included in the Catalog and the Student Handbook.

Exam policies. During quizzes, exams and the final all that you will need and will be allowed to use is a pen/pencil and the sheets of paper I give you. (all mobiles switched off , no textbooks, notes, calculators, etc.). You should work strictly by yourself – you should not communicate in any way with your classmates – violation of this will be considered cheating with all the ensuing consequences (see the AUBG documentation for the consequences of cheating). Cheating is not only talking to the person next to you (talking about anything: math, the problems, the weather, last nights party…) but also intentionally making your work available to others during the exam.

Assignments: I will assign sections from the textbook for you to read and learn for the next class (basicall, the material studied in class) and from time to time I will make reading pop-quizzes to check if you know the assigned part. I will also expect that on the average you spend 4hours per week (at least) on top of the regular Abstract Algebra classes working on problems from the book, or additional problems given in class. I will regularly give a list of problems as homework, I will not collect these homeworks but we will discuss the problemsduring the seminar or in class, volunteers will present solutions on the board.

Disclaimer: This syllabus is subject to modification. The instructor will communicate with students on any changes. The above distribution of weeks per chapter is only approximate.