Overview

The aim of this book is to teach the reader the topics in algebra which are useful in the study of computer science. In a clear, concise style, the author present the basic algebraic structures, and their applications to such topics as the finite Fourier transform, coding, complexity, and automata theory. The book can also be read profitably as a course in applied algebra for mathematics students.

Full Product Details

Author:   Lars Garding ,  Torbjörn Tambour ,  T Tambour ,  Betty Hannaford
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   Softcover reprint of the original 1st ed. 1988
Dimensions:   Width: 15.50cm , Height: 1.10cm , Length: 23.50cm
Weight:   0.330kg
ISBN:  

9780387967806

ISBN 10:   038796780
Pages:   198
Publication Date:   01 August 1988
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

1 Number theory.- 1.1 Divisibility.- 1.2 Congruences.- 1.3 The theorems of Fermat, Euler and Wilson.- 1.4 Squares and the quadratic reciprocity theorem.- 1.5 The Gaussian integers.- 1.6 Algebraic numbers.- 1.7 Appendix. Primitive elements and a theorem by Gauss.- Literature.- 2 Number theory and computing.- 2.1 The cost of arithmetic operations.- 2.2 Primes and factoring.- 2.3 Pseudo-random numbers.- Literature.- 3 Abstract algebra and modules.- 3.1 The four operations of arithmetic.- 3.2 Modules.- 3.3 Module morphisms. Kernels and images.- 3.4 The structure of finite modules.- 3.5 Appendix. Finitely generated modules.- Literature.- 4 The finite Fourier transform.- 4.1 Characters of modules.- 4.2 The finite Fourier transform.- 4.3 The finite Fourier transform and the quadratic reciprocity law.- 4.4 The fast Fourier transform.- Literature.- 5 Rings and fields.- 5.1 Definitions and simple examples.- 5.2 Modules over a ring. Ideals and morphisms.- 5.3 Abstract linear algebra.- Literature.- 6 Algebraic complexity theory.- 6.1 Polynomial rings in several variables.- 6.2 Complexity with respect to multiplication.- 6.3 Appendix. The fast Fourier transform is optimal.- Literature.- 7 Polynomial rings, algebraic fields, finite fields.- 7.1 Divisibility in a polynomial ring.- 7.2 Algebraic numbers and algebraic fields.- 7.3 Finite fields.- Literature.- 8 Shift registers and coding.- 8.1 The theory of shift registers.- 8.2 Generalities about coding.- 8.3 Cyclic codes.- 8.4 The BCH codes and the Reed-Solomon codes.- 8.5 Restrictions for error-correcting codes.- Literature.- 9 Groups.- 9.1 General theory.- 9.1.1 Groups and subgroups.- 9.1.2 Groups of bijections and normal subgroups.- 9.1.3 Groups acting on sets.- 9.2 Finite groups.- 9.2.1 Counting elements.- 9.2.2 Symmetry groups and the dihedral groups.- 9.2.3 The symmetric and alternating groups.- 9.2.4 Groups of low order.- 9.2.5 Applications of group theory to combinatorics.- Literature.- 10 Boolean algebra.- 10.1 Boolean algebras and rings.- 10.2 Finite Boolean algebras.- 10.3 Equivalence classes of switching functions.- Literature.- 11 Monoids, automata, languages.- 11.1 Matrices with elements in a non-commutative algebra.- 11.2 Monoids and languages.- 11.3 Automata and rational languages.- 11.4 Every rational language is accepted by a finite automaton.- Literature.- References.

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