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Overview
This book provides thorough coverage of the main topics of abstract algebra while offering nearly 100 pages of applications. A repetition and examples 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.
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From The Critics
For undergraduate students with little or no previous exposure to abstract mathematics, the textbook can be used for a onesemester course on groups and rings, a fullyear course, or a course designed by the instructor to emphasize certain aspects of algebra. The teaching principles are repetition, and examples before definitions. Annotation c. Book News, Inc., Portland, OR (booknews.com)Product Details
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Read an Excerpt
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 onesemester introductory course on groups and rings or for a fullyear 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 repetition and especially an examples first, definitions later order 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 selfcontained 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 A_{5}, 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 Galoistheoretic 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 onesemester 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 fullyear 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 Papantonopoulou The College of New Jersey aigli@tcnj.du
Table of Contents
Preface.
Acknowledgments
0. Background.
Sets and Maps. Equivalence Relations and Partitions. Properties of Z. Complex Numbers. Matrices.
A. GROUP THEORY.
1. Groups.
Examples and Basic Concepts. Subgroups. Cyclic Groups. Permutations.
2. Group Homomorphisms.
Cosets and Lagrange's Theorem. Homomorphisms. Normal Subgroups. Quotient Groups. Automorphisms.
3. Direct Products and Abelian Groups.
Examples and Definitions. Computing Orders. Direct Sums. Fundamental Theorem of Finite Abelian Groups.
4. Group Actions.
Group Actions and Cayley's Theorem. Stabilizers and Orbits in a Group Action. Burnside's Theorem and Applications. Conjugacy Classes and the Class Equation. Conjugacy in Sn and Simplicity of A5. The Sylow Theorems. Applications of the Sylow Theorems.
5. Composition Series.
Isomorphism Theorems. The JordanHölder Theorem. Solvable Groups.
B. RINGS AND FIELDS.
6. Rings.
Examples and Basic Concepts. Integral Domains. Fields.
7. Ring Homomorphisms.
Definitions and Basic Properties. Ideals. The Field of Quotients.
8. Rings of Polynomials.
Basic Concepts and Notation. The Division Algorithm in F[x]. More Applications of the Division Algorithm. Irreducible Polynomials. Cubic and Quartic Polynomials. Ideals in F[x]. Quotient Rings of F[x]. The Chinese Remainder Theorem for F[x].
9. Euclidean Domains.
Division Algorithms and Euclidean Domains. Unique Factorization Domains. Gaussian Integers.
10. Field Theory.
Vector Spaces. Algebraic Extensions. Splitting Fields. Finite Fields.
11. Geometric Constructions.
Constructible Real Numbers. Classical Problems. Constructions with Marked Ruler and Compass. Cubics and Quartics Revisited.
12. Galois Theory.
Galois Groups. The Fundamental Theorem of Galois Theory. Galois Groups of Polynomials. Geometric Constructions Revisited. Radical Extensions.
13. Historical Notes.
From Ahmes the Scribe to Omar Khayyam. From Gerolamo Cardano to C. F. Gauss. From Evariste Galois to Emmy Noether.
C. SELECTED TOPICS.
14. Symmetries.
Linear Transformations. Isometries. Symmetry Groups. Platonic Solids. Subgroups of the Special Orthogonal Group. Further Reading.
15. Grobner Bases.
Lexicographic Order. A Division Algorithm. Dickson's Lemma. The Hilbert Basis Theorem. Gröbner Bases and the Division Algorithm. Further Reading.
16. Coding Theory.
Linear Binary Codes. Error Correction and Coset Decoding. Standard Generator Matrices. The Syndrome Method. Cyclic Codes. Further Reading.
17. Boolean Algebras.
Lattices. Boolean Algebras. Circuits. Further Reading.
Answers and Hints to Selected Exercises.
Bibliography.
Index.
Preface
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 onesemester introductory course on groups and rings or for a fullyear 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 repetition and especially an examples first, definitions later order 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 selfcontained 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 A_{5}, 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 Galoistheoretic 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 onesemester 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 fullyear 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 Papantonopoulou
The College of New Jersey
aigli@tcnj.du