Riassunto
The mathematical concepts of abstract algebra may indeed be considered abstract, but its utility is quite concrete and continues to grow in importance. Unfortunately, the practical application of abstract algebra typically involves extensive and cumbersome calculations-often frustrating even the most dedicated attempts to appreciate and employ its intricacies. Now, however, sophisticated mathematical software packages help obviate the need for heavy number-crunching and make fields dependent on the algebra more interesting-and more accessible. Applications of Abstract Algebra with Maple opens the door to cryptography, coding, Polya counting theory, and the many other areas dependent on abstract algebra. The authors have carefully integrated Maple V throughout the text, enabling readers to see realistic examples of the topics discussed without struggling with the computations. But the book stands well on its own if the reader does not have access to the software. The text includes a first-chapter review of the mathematics required-groups, rings, and finite fields-and a Maple tutorial in the appendix along with detailed treatments of coding, cryptography, and Polya theory applications. Applications of Abstract Algebra with Maple packs a double punch for those interested in beginning-or advancing-careers related to the applications of abstract algebra. It not only provides an in-depth introduction to the fascinating, real-world problems to which the algebra applies, it offers readers the opportunity to gain experience in using one of the leading and most respected mathematical software packages available.
Contenuti
Preliminary Mathematics
Permutation Groups
Cosets and Quotient Groups
Rings and Euclidean Domains
Finite Fields
Finite Fields with Maple
The Euclidean Algorithm
Block Designs
General Properties of Block Designs
Hadamard Matrices
Hadamard Matrices with Maple
Difference Sets
Difference Sets with Maple
Error-Correcting Codes
General Properties of Codes
Hadamard Codes
Reed-Muller Codes
Reed-Muller Codes with Maple
Linear Codes
Hamming Codes with Maple
BCH Codes
Construction of BCH Codes
Error Correction in BCH Coides
BCH Codes with Maple
Reed-Solomon Codes
Construction of Reed-Solomon Codes
Error Correction in Reed-Solomon Codes
Proof of Reed-Solomon Error Correction
Binary Reed-Solomon Codes
Reed-Solomon Codes with Maple
Reed-Solomon Codes in Voyager 2
Algebraic Cryptography
Some elementary Cryptosystems
The Hill Cryptosystem
The Hill Cryptosystem with Maple
Generalizations of the Hill Cryptosystem
The Two-Message Problem
The RSA Cryptosystem
Mathematical Prerequisites
RSA Encryption and Decryption
The RSA Cryptosystem with Maple
A Note on Modular Exponentiation
A Note on Primality Testing
A Note on Integer Factorization
A Note on Digital Signatures
The Diffie-Hellman Key Exchange
Elliptic Curve Cryptography
The ElGamal Cryptosystem
The ElGamal Crytosystem with Maple
Elliptic Curves
Elliptic Curves with Maple
Elliptic Curve Cryptography
Elliptic Curve Cryptography with Maple
Polya Theory
Group Actions
Burnside's Theorem
The Cycle Index
The Pattern Inventory
The Pattern Inventory with Maple
Switching Functions
Switching Functions with Maple
Appendices
Basic Maple Tutorial
Some Maple Linear Algebra Commands
User-Written Maple Procedures
Bibliography
Hints and Solutions to Selected Written Exercises
Index
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