Abstract Algebra: An Interactive Approach(Textbooks in Mathematics)

By integrating the use of GAP and Mathematica(R), Abstract Algebra: An Interactive Approach presents a hands-on approach to learning about groups, rings, and fields. Each chapter includes both GAP and Mathematica commands, corresponding Mathematica notebooks, traditional exercises, and several interactive computer problems that utilize GAP and Mathematica to explore groups and rings. Although the book gives the option to use technology in the classroom, it does not sacrifice mathematical rigor. It covers classical proofs, such as Abel's theorem, as well as many graduate-level topics not found in most standard introductory texts. The author explores semi-direct products, polycyclic groups, Rubik's Cube(R)-like puzzles, and Wedderburn's theorem. He also incorporates problem sequences that allow students to delve into interesting topics in depth, including Fermat's two square theorem. 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.

Table of Contents:
Understanding the Group Concept Introduction to Groups Modular Arithmetic Prime Factorizations The Definition of a Group The Structure within a Group Generators of Groups Defining Finite Groups in Mathematica and GAP Subgroups Patterns within the Cosets of Groups Left and Right Cosets How to Write a Secret Message 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 Pyraminx™ Introduction to Rings Groups with an Additional Operation The Definition of a Ring Entering Finite Rings into GAP and Mathematica 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 Mathematica or GAP 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 Solutions of Polynomial Equations Using Radicals Bibliography Answers to Odd Problems Index Problems appear at the end of each chapter.

About the Author :
William Paulsen is a Professor of Mathematics at Arkansas State University.

Review :
"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 1173