ebook img

Applications of Abstract Algebra PDF

200 Pages·1985·7.175 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Applications of Abstract Algebra

Auplications OS Oe V 4fP) s iS Elon College, North Carolina pelea aeOi sae NO LONGER THE PR OPE RIY OF EMON UNIVERSITY LIBRARY 8638596 “> Anplications of Abstract Algebra George Mackiw John Wiley & Sons New York / Chichester / Brisbane / Toronto / Singapore 863596 Copyright (c)1985 by John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. ISBN 0-471-81078-9 Printed in the United States of America 10 ORS sOrar aes 2s 1 Preface Abstract or modern algebra is considered by many to be one of the more beautiful, elegant and coherent branches of mathematics. Yet, in spite of the greater emphasis today on the use of discrete methods, the study of groups, rings and fields seems to have suffered. A primary reason may be that traditional presentations of the subject have not placed much emphasis on motivation and applications, providing perhaps less reason for the average student to be attracted to or to study the subject. For example, while it is often mentioned that symmetry groups are important in chemistry and physics, rarely are concrete examples of such groups given in introductory texts. The goal of this text is to provide a collection of applications of abstract algebra to a student who is currently taking a One or two semester introduction to the basics of groups, rings and fields. The text is thus intended for use as a supplement to an Abstract Algebra course, though it certainly may also be used in a separate course on applications or as a reference. The applications included here cover exact computing, error-correcting codes, the construction of block designs useful in statistics, crystallography, integer programming, cryptography and combinatorics. A more detailed description of individual chapters follows below. My motivation in writing this text was to present a series of applications that would be accessible to the student in a first course, provide examples of algebra in use, contain topics not previously gathered together in text form, and hopefully be interesting. Topics such as classical Greek constructions and applications of Galois Theory have deliberately been omitted since these are precisely the ones that are covered in many excellent texts. Applications to quantum mechanics and physics are also absent since they tend to require a background deeper than that possessed by the beginning student. The chapters that follow are, with two exceptions, independent of each other. We include a description of the content and prerequisites for each chapter. 1) Computing in Zu - Exact Solutions of Linear Equations Problems of round-off and precision are inherent in the floating-point arithmetic used in computers. This chapter describes how the arithmetic of the ring Zy and the Chinese Remainder Theorem may be helpful in avoiding such problems when dealing with systems of linear equations. Prerequisites: elementary linear algebra, modular arithmetic, ring homomorphisms. 2) Block Designs The use of abelian groups, finite fields and rings in the construction of block designs, structures which are useful in the design of experiments. Prerequisites: elementary theory of abelian groups, modular arithmetic, the fields GF(p"), primitive elements. 3) Error Correcting Codes I: Hamming Codes An introduction to the basics of error correcting codes. This chapter features a detailed development of the Hamming codes and the rationale behind their error correcting capabilities. Prerequisites: elementary linear algebra, the field GF(2), cosets, the fundamental theorem of group homomorphisms. 4) Error Correcting Codes II: BCH Codes This requires the previous chapter and features the development of the multiple error correcting BCH codes. It includes a step by step exposition of the error correcting procedures attached to BCH codes, which serve as a wonderful primer for reinforcing the arithmetic of finite fields. Prerequisites: elementary linear algebra, the finite fields GF(2"), primitive elements, minimal polynominals. 5) Crystallographic Groups in the Plane I This serves as an introduction to the classification of crystals via their symmetry groups. Introduces the notion of crystallographic space groups, translation subgroups and point groups in two dimensions. Prerequisites: elementary linear algebra and group theory, quotient groups, fundamental theorem of group homomorphisms. 6) Appendix: Orthogonal Matrices in Two Dimensions Develops from scratch results on 2 x 2 orthogonal matrices which are needed in the above chapter. Prerequisites: familiarity with 2 x 2 matrices and determinants. 7) Crystallographic Groups in the Plane II This is a continuation of the previous chapter and provides a rigorous development of the fact that there are only 17 inequivalent crystallographic groups in the plane. These are illustrated by concrete matrix representations and drawings. Prerequisites: “same as the previous chapter. 8) The RSA Public Key Cryptosystem Discusses the issues behind modern public key cryptography and develops the algebra underlying the Rivest-Shamir-Adleman system. Includes a discussion of recent developments in primality testing and factorization. Prerequisites: modular arithmetic, the ring Zy, Lagrange's Theorem for groups. 9) Integer Programming Introduces integer programming through optimization problems. Develops the role finite abelian groups play in the solution of integer programming problems. Prerequisites: elementary linear algebra and linear programming, use of the simplex algorithm, group homomorphisms, quotient groups, the exponent of a finite abelian group. 10) Group Theory and Counting The role group theory plays in combinatorial problems; includes a derivation of Burnside's Lemma with many concrete examples. Prerequisites: elementary group theory, cosets, equivalence relations. As may be evident, elementary linear algebra is invoked throughout. This should present no significant difficulty, since it is now commonplace for most students to encounter linear algebra in either freshman or sophomore year. With the exception of the two chapters each on crystallography and error correcting codes, the individual sections of the text are independent of each other and may be used in any order desired. Each chapter includes a set of exercises and a list of further references. I should like to thank the staff of the Loyola College Communication Center, especially Charlette Hinchliffe, Nancy Marshall, Mary Muhler, and Marion Weilgosz for their diligent work in preparing this manuscript. Dean David Roswell was very helpful in his support and encouragement. It was my pleasure to engage in many interesting conversations with Professor Alan Goldman of Johns Hopkins University during a sabbatical year spent at the Center for Applied Mathematics at the National Bureau of Standards. It was he who provided the idea for the chapter on integer programming. It is my hope that both students and teachers of abstract algebra will find this collection useful. George Mackiw Baltimore, Maryland & > Cia ba ‘Caine tl! Poe enue ts ares 5 Stina) Se ga Cam Braz pews

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.