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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