CiNii Books - Applications of abstract algebra with Maple

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.

Table of Contents

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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Applications of abstract algebra with Maple

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