9780130882547

Algebra Pure and Applied

by
  • ISBN13:

    9780130882547

  • ISBN10:

    0130882542

  • Edition: 1st
  • Format: Paperback
  • Copyright: 2001-05-24
  • Publisher: Pearson

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Summary

This book provides thorough coverage of the main topics of abstract algebra while offering nearly 100 pages of applications. Arepetition andexamples first approach introduces learners to mathematical rigor and abstraction while teaching them the basic notions and results of modern algebra. Chapter topics include group theory, direct products and Abelian groups, rings and fields, geometric constructions, historical notes, symmetries, and coding theory. For future teachers of algebra and geometry at the high school level.

Table of Contents

Preface xiii
Acknowledgments xvii
Background
1(38)
Sets and Maps
1(6)
Equivalence Relations and Partitions
7(3)
Properties of Z
10(14)
Complex Numbers
24(7)
Matrices
31(8)
Part A Group Theory
Groups
39(35)
Examples and Basic Concepts
39(11)
Subgroups
50(6)
Cyclic Groups
56(6)
Permutations
62(12)
Group Homomorphisms
74(34)
Cosets and Lagrange's Theorem
74(7)
Homomorphisms
81(7)
Normal Subgroups
88(6)
Quotient Groups
94(9)
Automorphisms
103(5)
Direct Products and Abelian Groups
108(23)
Examples and Definitions
108(5)
Computing Orders
113(3)
Direct Sums
116(6)
Fundamental Theorem of Finite Abelian Groups
122(9)
Group Actions
131(41)
Group Actions and Cayley's Theorem
131(6)
Stabilizers and Orbits in a Group Action
137(5)
Burnside's Theorem and Applications
142(6)
Conjugacy Classes and the Class Equation
148(5)
Conjugacy in Sn and Simplicity of A5
153(6)
The Sylow Theorems
159(6)
Applications of the Sylow Theorems
165(7)
Composition Series
172(21)
Isomorphism Theorems
172(6)
The Jordan-Holder Theorem
178(6)
Solvable Groups
184(9)
Part B Rings and Fields
Rings
193(17)
Examples and Basic Concepts
193(6)
Integral Domains
199(3)
Fields
202(8)
Ring Homomorphisms
210(22)
Definitions and Basic Properties
210(5)
Ideals
215(9)
The Field of Quotients
224(8)
Rings of Polynomials
232(47)
Basic Concepts and Notation
232(7)
The Division Algorithm in F[x]
239(6)
More Applications of the Division Algorithm
245(5)
Irreducible Polynomials
250(10)
Cubic and Quartic Polynomials
260(6)
Ideals in F[x]
266(3)
Quotient Rings of F[x]
269(4)
The Chinese Remainder Theorem for F[x]
273(6)
Euclidean Domains
279(20)
Division Algorithms and Euclidean Domains
279(7)
Unique Factorization Domains
286(8)
Gaussian Integers
294(5)
Field Theory
299(42)
Vector Spaces
299(9)
Algebraic Extensions
308(14)
Splitting Fields
322(12)
Finite Fields
334(7)
Geometric Constructions
341(30)
Constructible Real Numbers
341(11)
Classical Problems
352(5)
Constructions with Marked Ruler and Compass
357(6)
Cubics and Quartics Revisited
363(8)
Galois Theory
371(62)
Galois Groups
371(12)
The Fundamental Theorem of Galois Theory
383(15)
Galois Groups of Polynomials
398(17)
Geometric Constructions Revisited
415(8)
Radical Extensions
423(10)
Historical Notes
433(20)
From Ahmes the Scribe to Omar Khayyam
433(6)
From Gerolamo Cardano to C. F. Gauss
439(6)
From Evariste Galois to Emmy Noether
445(8)
Part C Selected Topics
Symmetries
453(22)
Linear Transformations
453(6)
Isometries
459(5)
Symmetry Groups
464(2)
Platonic Solids
466(2)
Subgroups of the Special Orthogonal Group
468(7)
Further Reading
472(3)
Grobner Bases
475(15)
Lexicographic Order
475(3)
A Division Algorithm
478(4)
Dickson's Lemma
482(3)
The Hilbert Basis Theorem
485(1)
Grobner Bases and the Division Algorithm
486(4)
Further Reading
488(2)
Coding Theory
490(17)
Linear Binary Codes
490(4)
Error Correction and Coset Decoding
494(3)
Standard Generator Matrices
497(2)
The Syndrome Method
499(6)
Cyclic Codes
505(2)
Further Reading
502(5)
Boolean Algebras
507(18)
Lattices
507(4)
Boolean Algebras
511(6)
Circuits
517(8)
Further Reading
520(5)
Answers and Hints to Selected Exercises 525(14)
Bibliography 539(2)
Index 541

Excerpts

This book aims to provide thorough coverage of the main topics of abstract algebra while remaining accessible to students with little or no previous exposure to abstract mathematics. It can be used either for a one-semester introductory course on groups and rings or for a full-year course. More specifics on possible course plans using the book are given in this preface. Style of Presentation Over many years of teaching abstract algebra to mixed groups of undergraduates, including mathematics majors, mathematics education majors, and computer science majors, I have become increasingly aware of the difficulties students encounter making their first acquaintance with abstract mathematics through the study of algebra. This book, based on my lecture notes, incorporates the ideas I have developed over years of teaching experience on how best to introduce students to mathematical rigor and abstraction while at the same time teaching them the basic notions and results of modern algebra. Two features of the teaching style I have found effective are repetitionand especially an examples first, definitions laterorder of presentation. In this book, as in my lecturing, the hard conceptual steps are always prepared for by working out concrete examples first, before taking up rigorous definitions and abstract proofs. Absorption of abstract concepts and arguments is always facilitated by first building up the student''s intuition through experience with specific cases. Another principle that is adhered to consistently throughout the main body of the book (Parts A and B) is that every algebraic theorem mentioned is given either with a complete proof, or with a proof broken up into to steps that the student can easily fill in, without recourse to outside references. The book aims to provide a self-contained treatment of the main topics of algebra, introducing them in such a way that the student can follow the arguments of a proof without needing to turn to other works for help. Throughout the book all the examples, definitions, and theorems are consecutively numbered in order to make locating any particular item easier for the reader. Coverage of Topics In order to accommodate students of varying mathematical, backgrounds, an optional Chapter 0, at the beginning, collects basic material used in the development of the main theories of algebra. Included are, among other topics, equivalence relations, the binomial theorem, De Moivre''s formula for complex numbers, and the fundamental theorem of arithmetic. This chapter can be included as part of an introductory course or simply referred to as needed in later chapters. Special effort is made in Chapter 1 to introduce at the beginning all main types of groups the student will be working with in later chapters. The first section of the chapter emphasizes the fact that concrete examples of groups come from different sources, such as geometry, number theory, and the theory of equations. Chapter 2 introduces the notion of group homomorphism first and then proceeds to the study of normal subgroups and quotient groups. Studying the properties of the kernel of a homomorphism before introducing the definition of a normal subgroup makes the latter notion less mysterious for the student and easier to absorb and appreciate. A similar order of exposition is adopted in connection with rings. After the basic notion of a ring is introduced in Chapter 6, Chapter 7 begins with ring homomorphisms, after which consideration of the properties of the kernels of such homomorphisms gives rise naturally to the notion of an ideal in a ring. Each chapter is designed around some central unifying theme. For instance, in Chapter 4 the concept of group action is used to unify such results as Cayley''s theorem, Burnside''s counting formula, the simplicy of A5, and the Sylow theorems and their applications. The ring of polynomials over a field is the central topic of Part B, Rings and Fields, and is given a full chapter of its own, Chapter 8. The traditional main topic in algebra, the solution of polynomial equations, is emphasized. The solutions of cubics and quartics are introduced in Chapter 8. In Chapter 9 Euclidean domains and unique factorization domains are studied, with a section devoted to the Gaussian integers. The fundamental theorem of algebra is stated in Chapter 10. In Chapter 11 the connection among solutions of quadratic, cubic, and quartic polynomial equations and geometric constructions is explored. In Chapter 12, after Galois theory is developed, it is applied to give a deeper understanding of all these topics. For instance, the possible Galois groups of cubic and quartic polynomials are fully worked out, and Artin''s Galois-theoretic proof of the fundamental- theorem of algebra, using nothing from analysis but the intermediate value theorem, is presented. The chapter, and with it the main body of the book, culminates in the proof of the insolubility of the general quintic and the construction of specific examples of quintics that are not solvable by radicals. A brief history of algebra is given in Chapter 13, after Galois theory (which was the main historical source of the group concept) has been treated, thus making a more meaningful discussion of the evolution of the subject possible. A collection of additional topics, several of them computational, is provided in Part C. In contrast to the main body of the book (Parts A and B), where completeness is the goal, the aim in Part C is to give the student an introduction to--and some taste of--a topic, after which a list of further references is provided for those who wish to learn more. Instructors may include as much or as little of the material on a given topic as time and inclination indicate. Each chapter in the book is divided into sections, and each section provided with a set of exercises, beginning with the more computational and proceeding to the more theoretical. Some of the theoretical exercises give a first introduction to topics that will be treated in more detail later in the book, while others introduce supplementary topic not otherwise covered, such as Cayley digraphs, formal power series, and the existence of transcendental numbers. Suggestions for Use A one-semester introductory course on groups and rings might include Chapter 0 (optional); Chapters 1, 2, and 3 on groups; and Chapters 6, 7, and 8 on rings. For a full-year course, Parts A and B, Chapters 1 through 12, offer a comprehensive treatment of the subject. Chapter 9, on Euclidean domains, and Chapter 11, on geometric constructions, can be treated as optional supplementary topics, depending on time arid the interest of the students and the instructor. An instructor''s manual, with solutions to all exercises plus further comments and suggestions, is available. Instructors can obtain it by directly contacting the publisher, Prentice Hall. Acknowledgments It is a pleasure to acknowledge various contributors to the development of this book. First I should thank the students of The College of New Jersey who have taken courses based on a first draft. I am grateful also to my colleagues Andrew Clifford, Tom Hagedorn, and Dave Reimer for useful suggestions. Special thanks are due to my colleague Ed Conjura, who taught from a craft of the book and made invaluable suggestions for improvement that have been incorporated into the final version. I am also most appreciative of the efforts of the anonymous referees engaged by the publisher, who provided many helpful and encouraging comments. My final word of gratitude goes to my family--to my husband, John Burgess, and to our sons, Alexi and Fokion--for their continuous understanding and support throughout the preparation of the manuscript. Aigli Papan

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