Modern Algebra    MATH 205

Assisted

Dr. Daniel Reich

Course:

Modern Algebra

Math 205, section 1 (CRN 059278)

Barton Hall, room 202

Tues, Thurs 1:10 - 2:30

Textbook:

Algebra Pure & Applied, by Aigli Papantonopoulou
(Prentice Hall, 2002)

 

This is an introductory course in abstract algebra for students interested in mathematics and the teaching of mathematics.

Prerequisites

Linear algebra (Math 147 or 148) or permission of the instructor

Concepts that underly the subject:

1. Elementary set theory, including Cartesian products, equivalence relations, mappings
2. Techniques of proof, including mathematical induction
3. Properties of the integers, including divisibility, greatest common divisor, primes and prime factorization, congruence modulo n.
4. Real and complex numbers, matrices

Most of these topics are reviewed in Chapter 0 of the text.

The course covers most of the material in the first six chapters of the book. There are lots of problems, both easy and hard, at the end of each section. A large part of your grade will be based on your written solutions to many of these problems. There will also be a mid-term exam and a final, each based on half of the course. Occasional quizzes and other assignments will be sporadic and spontaneous, and not always announced in advance. Your participation in the classroom discussion will also contribute to your grade.

My office hours are tentatively scheduled as shown above. I hope you will come to ask questions or to talk to me about any ideas you find either confusing or especially interesting. Please feel free to talk to me about the course anytime you find me in my office, not just during the scheduled hours.

Overview of the topics covered Chapter 1 - After some preliminary discussion of the language of set theory, numbers and matrices (Chapter 0), we introduce the concept of a group, and some of the elementary examples and properties of groups and group elements: subgroups, orders, cyclic groups. Chapter 2 - We next discuss some of the fundamental structures of the theory of groups: homomorphisms, cosets, normal subgroups and factor groups. Some examples and applications are mixed in. Chapter 3 - We study direct products and sums, and the fundamental theorem of finite abelian groups. Chapter 6 - At this point, we switch our focus to rings and the associated structures: integral domains and fields, Chapter 7 - We study ideals and factor rings, homomorphisms, fields of quotients. Chapter 8 - These ideas are applied in the context of rings of polynomials, a very familiar setting.
If there is time: Chapter 9 - This discussion is generalized to "integral domains", a type of ring which shares many of the properties of rings of integers and polynomials. Chapter 10 - We introduce fields, rings like the rationals or real numbers, in which the full strength of ordinary arithmetical operations is available.

Sections to be covered:

Chapter 0: Preliminaries

Section 1 - Sets and Maps

Section 2 - Equivalence relations & partitions

Section 3, 4, 5 - Properties of integers, complex numbers, matrices

Chapter 1: Groups

Section 1 - Examples and basic concepts

Section 2 - Subgroups

Section 3 - Cyclic Groups

Section 4 - Permutations

Chapter 2: Group Homomorphisms

Section 1 - Cosets and Lagrange's Theorem

Section 2 - Homomorphisms

Section 3 - Normal subgroups

Section 4 - Quotient Groups

Section 5 - Automorphisms

Chapter 3: Direct Products and Abelian Groups

Section 1 - Examples and Definitions

Section 2 - Computing Orders

Section 3 - Direct Sums

Section 4 - Fundamental Theorem of Finite Abelian Groups

Chapter 6: Rings

Section 1 - Examples and basic concepts

Section 2 - Integral domains

Section 3 - Fields

Chapter 7: Ring Homomorphisms

Section 1 - Definitions and Basic Properties

Section 2 - Ideals

Section 3 - The Field of Quotients

Chapter 8: Rings of Polynomials

Section 1 - Basic Concepts and Notation

Section 2 - The Division Algorithm in Fx

Section 3 - More Applications of the Division Algorithm

Section 4 - Irreducible Polynomials

Section 5 - Cubic and Quartic Polynomials

Section 6 - Ideals in Fx

Section 7 - Quotient Rings of Fx

Section 8 - The Chinese Remainder Theorem for Fx

If there is time:
Chapter 9: Euclidean Domains

Section 1 - Division Algorithms and Euclidean Domains

Section 2 - Unique Factorization Domains

Section 3 - Gaussian Integers

Chapter 10: Field Theory

Section 1 - Vector spaces

Section 2 - Algebraic extensions

Section 3 - Splitting fields

Section 4 - Finite fields

 

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