2nd Edition
Abstract Algebra An Interactive Approach, Second Edition
The new edition of Abstract Algebra: An Interactive Approach presents a hands-on and traditional approach to learning groups, rings, and fields. It then goes further to offer optional technology use to create opportunities for interactive learning and computer use.
This new edition offers a more traditional approach offering additional topics to the primary syllabus placed after primary topics are covered. This creates a more natural flow to the order of the subjects presented. This edition is transformed by historical notes and better explanations of why topics are covered.
This innovative textbook shows how students can better grasp difficult algebraic concepts through the use of computer programs. It encourages students to experiment with various applications of abstract algebra, thereby obtaining a real-world perspective of this area.
Each chapter includes, corresponding Sage notebooks, traditional exercises, and several interactive computer problems that utilize Sage and Mathematica® to explore groups, rings, fields and additional topics.
This text does not sacrifice mathematical rigor. It covers classical proofs, such as Abel’s theorem, as well as many topics not found in most standard introductory texts. The author explores semi-direct products, polycyclic groups, Rubik’s Cube®-like puzzles, and Wedderburn’s theorem. The author also incorporates problem sequences that allow students to delve into interesting topics, including Fermat’s two square theorem.
Preliminaries
Integer Factorization
Functions
Modular Arithmetic
Rational and Real Numbers
Understanding the Group Concept
Introduction to Groups
Modular Congruence
The Definition of a Group
The Structure within a Group
Generators of Groups
Defining Finite Groups in Sage
Subgroups
Patterns within the Cosets of Groups
Left and Right Cosets
Writing Secret Messages
Normal Subgroups
Quotient Groups
Mappings between Groups
Isomorphisms
Homomorphisms
The Three Isomorphism Theorems
Permutation Groups
Symmetric Groups
Cycles
Cayley's Theorem
Numbering the Permutations
Building Larger Groups from Smaller Groups
The Direct Product
The Fundamental Theorem of Finite Abelian Groups
Automorphisms
Semi-Direct Products
The Search for Normal Subgroups
The Center of a Group
The Normalizer and Normal Closure Subgroups
Conjugacy Classes and Simple Groups
The Class Equation and Sylow's Theorems
Solvable and Insoluble Groups
Subnormal Series and the Jordan-Hölder Theorem
Derived Group Series
Polycyclic Groups
Solving the PyraminxTM
Introduction to Rings
The Definition of a Ring
Entering Finite Rings into Sage
Some Properties of Rings
The Structure within Rings
Subrings
Quotient Rings and Ideals
Ring Isomorphisms
Homomorphisms and Kernels
Integral Domains and Fields
Polynomial Rings
The Field of Quotients
Complex Numbers
Ordered Commutative Rings
Unique Factorization
Factorization of Polynomials
Unique Factorization Domains
Principal Ideal Domains
Euclidean Domains
Finite Division Rings
Entering Finite Fields in Sage
Properties of Finite Fields
Cyclotomic Polynomials
Finite Skew Fields
The Theory of Fields
Vector Spaces
Extension Fields
Splitting Fields
Galois Theory
The Galois Group of an Extension Field
The Galois Group of a Polynomial in Q
The Fundamental Theorem of Galois Theory
Applications of Galois Theory
Appendix: Sage vs. Mathematica®
Answers to Odd-Numbered Problems
Bibliography
Biography
William Paulsen, PhD, professor of mathematics, Arkansas State University, USA
Praise for previous editions:
"The textbook gives an introduction to algebra. The course includes the explanation on how to use the computer algebra systems GAP and Mathematica …The book can be used for an undergraduate-level course (chapter 1-4 and 9-12) or a second semester graduate-level course."
—Gerhard Pfister, Zentralblatt MATH
