Interactive Abstract Algebra
Table of Contents
The book "Interactive Abstract Algebra" has two parts--the first dealing
with group theory, and the second working with rings and fields culminating
with Galois theory. These are intended to be used in a two semester
Modern Algebra course. But each chapter of the book has a corresponding
Mathematica notebook, designed to enhance student learning.
Table of Contents
Group01.mb -- Understanding the Group Concept
- Terry the Dancing Triangle
- Modular Arithmetic
- Prime Factorizations
- The Definition of a Group
Group02.mb -- The Structure Within a Group
- Generators of Groups
- Defining Finite Groups in Mathematica
- Subgroups
Group03.mb -- Patterns Within the Cosets of Groups
- Left and Right Cosets
- How to Write a Secret Message
- Normal Subgroups
- Quotient Groups
Group04.mb -- Mappings Between Groups
- Isomorphisms
- Homomorphisms
- The Three Isomorphism Theorems
Group05.mb -- Permutation Groups
- Symmetric Groups
- Caylay's Theorem
- Cycles
- Numeration of the Permutations
Group06.mb -- Building Larger Groups from Smaller Groups
- The Direct Product
- Fundamental Theorem of Finite Abelian Groups
- Automorphisms
- Semi-direct Products
Group07.mb -- The Search for Normal Subgroups
- The Center of a Group
- The Normalizer and Generated Normal Groups
- Conjugacy Classes and Simple Groups
- The Class Equation and Sylow's Theorems
Group08.mb -- Solvable and Insoluble Groups
- Subnormal Series and the Jordan-Hölder Theorem
- Derived Group Series
- Representing Finite Solvable Groups with Mathematica
- Solving the Pyramix™
Part 2: Ring theory:
Ring09.mb -- Introduction to Rings
- Some Groups with an Additional Operation
- Definition of a Ring
- Entering Finite Rings into MATHEMATICA
- Some Properties of Rings
Ring10.mb -- The Structure Within Rings
- Subrings
- Quotient Rings and Ideals
- Ring Isomorphisms
- Homomorphisms and Kernals
Ring11.mb -- Integral Domains and Fields
- Polynomial Rings
- The Field of Quotients
- Ordered Commutative Rings
- Complex Numbers
Ring12.mb -- Unique Factorization
- Factorization of Polynomials
- Unique Factorization Domains
- Principal Ideal Domains
- Euclidean Domains
Ring13.mb -- Finite Division Rings
- Entering Finite Fields in MATHEMATICA
- Properies of Finite Fields
- Cyclitomic Polynomials
- Finite Skew Fields
Ring14.mb -- The Theory of Fields
- Extension Fields
- Splitting Fields
- Special Properties of Splitting Fields
Ring15.mb -- Galois Theory