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 S_n and Simplicity of A_n

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

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