Modern Algebra   MATH 305

Assisted:

Dr. Daniel Reich

Course:
Modern Algebra
Math 305, section 1 (CRN 059278)
Curtis Hall, room 303
Tues, Thurs 1:10 - 2:30
Textbook:
Algebra Pure & Applied, by Aigli Papantonopoulou
(Prentice Hall, 2002)
 

This is a second course in abstract algebra for students interested in mathematics and the teaching of mathematics.

Prerequisites
One semester of modern algebra (Math 205 or the equivalent) or permission of the instructor
Concepts that underly the subject:
  1. Elementary set theory, including Cartesian products, equivalence relations, mappings
  2. Elementary number theory, including induction, integral divisibility, GCD's, primes.
  3. The introductory theory of groups, rings and fields.
These topics are in Chapters 1-3 and 6-8 of the text.
The course covers most of the material in Chapters 4, 5 and 8-12 of the text. Additional selected topics from chapter 14-17 will be covered if time allows.

A large part of your grade will be based on your written solutions to assigned 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 4 - We study group actions and associated structures such as orbits and stabilizers. The we specialize to the important case of conjugacy within a group, and develop the class equation. This leads to the crucial Sylow theorems. Chapter 5 - Composition series: a group may be studied by breaking it up into a sequence of simple "extensions". Chapters 8 & 9 - We return to ring theory in order to complete our survey of polynomial rings, and to look at the more general concept of a euclidean domain. Chapter 10 - At this point, we switch our focus to fields: algebraic extensions, splitting fields and as a special case of interest, finite fields. Chapter 11 - Basic field theory is used to analyze some classical questions about geometric ruler and compass construction. Chapter 12 - The principal goal of the course: Galois theory.
If there is time: Additional topics to be chosen later.

Sections to be covered:

Chapter 4: Group Actions
Section 1 - Group Actions and Cayley's Theorem
Section 2 - Stabilizers and Orbits in a Group Action
Section 4 - Conjugacy Classes and the Class Equation
Section 5 - Conjugacy in Sn and Simplicity of An
Section 6 - The Sylow Theorems
Section 7 - Applications of the Sylow Theorems
Chapter 5: Composition Series
Section 1 - Isomorphism Theorems
Section 2 - The Jordan-Hder Theorem
Section 3 - Solvable Groups
Chapter 8: Rings of Polynomials
Section 4 - Irreducible Polynomials
Section 5 - Cubic and Quartic Polynomials
Section 6 - Ideals in F[x]
Section 7 - Quotient Rings of F[x]
Section 8 - The Chinese Remainder Theorem for F[x]
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
Chapter 11: Geometric Constructions
Section 1 - Constructible Real Numbers
Section 2 - Classical Problems
Section 3 - Construction with Marked Ruler and Compass
Chapter 12: Galois Theory
Section 1 - Galois Groups
Section 2 - The Fundamental Theorem of Galois Theory
Section 3 - Galois Groups of Polynomials
Section 4 - Geometric Construction Revisited
Section 5 - Radical Extensions

 

Printer Friendly Version

Back Home