Abstract Algebra: An Interactive Approach (Textbooks in Mathematics 40 2nd Revised edition)
By: William Paulsen (author)Hardback
Only 1 in stock
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(R) 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(R)-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.
William Paulsen, PhD, professor of mathematics, Arkansas State University, USA
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-Holder 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(R) Answers to Odd-Numbered Problems Bibliography
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- ID: 9781498719766
2nd Revised edition
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