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Contemporary Abstract Algebra 10th edition [Hardback]

4.05/5 (451 ratings by Goodreads)
  • Formāts: Hardback, 654 pages, height x width: 234x156 mm, weight: 1264 g, 50 Illustrations, black and white
  • Sērija : Textbooks in Mathematics
  • Izdošanas datums: 30-Dec-2020
  • Izdevniecība: Chapman & Hall/CRC
  • ISBN-10: 0367651785
  • ISBN-13: 9780367651787
Citas grāmatas par šo tēmu:
  • Formāts: Hardback, 654 pages, height x width: 234x156 mm, weight: 1264 g, 50 Illustrations, black and white
  • Sērija : Textbooks in Mathematics
  • Izdošanas datums: 30-Dec-2020
  • Izdevniecība: Chapman & Hall/CRC
  • ISBN-10: 0367651785
  • ISBN-13: 9780367651787
Citas grāmatas par šo tēmu:
Contemporary Abstract Algebra, Tenth Edition

For more than three decades, this classic text has been widely appreciated by instructors and students alike. The book offers an enjoyable read and conveys and develops enthusiasm for the beauty of the topics presented. It is comprehensive, lively, and engaging.

The author presents the concepts and methodologies of contemporary abstract algebra as used by working mathematicians, computer scientists, physicists, and chemists. Students will learn how to do computations and to write proofs. A unique feature of the book are exercises that build the skill of generalizing, a skill that students should develop but rarely do. Applications are included to illustrate the utility of the abstract concepts.

Examples and exercises are the heart of the book. Examples elucidate the definitions, theorems, and proof techniques; exercises facilitate understanding, provide insight, and develop the ability to do proofs. The exercises often foreshadow definitions, concepts, and theorems to come.

Changes for the tenth edition include new exercises, new examples, new quotes, and a freshening of the discussion portions. The hallmark features of previous editions of the book are enhanced in this edition. These include:





A good mixture of approximately 1900 computational and theoretical exercises, including computer exercises, that synthesize concepts from multiple chapters Approximately 300 worked-out examples from routine computations to the challenging Many applications from scientific and computing fields and everyday life Historical notes and biographies that spotlight people and events Motivational and humorous quotations Numerous connections to number theory and geometry

While many partial solutions and sketches for the odd-numbered exercises appear in the book, an Instructors Solutions Manual written by the author has comprehensive solutions for all exercises and some alternative solutions to develop a critical thought and deeper understanding. It is available from CRC Press only. The Student Solution Manual has comprehensive solutions for all odd-numbered exercises and many even-numbered exercises.

Recenzijas

"It has now been 35 years since Gallian's classic textbook Contemporary Abstract Algebra was first published. The book is deservedly popular with instructors of abstract algebra courses. It is written at an appropriate level for junior and senior undergraduates, has lucid coverage of all of the standard topics and several nonstandard ones (Frieze Groups and Crystallographic Groups, Coding Theory, Greek Geometric Construction Problems, etc), is example-driven, and contains thousands of exercises at various levels of difficulty. Moreover, every chapter begins with an interesting quote or two (by as diverse a group of people as Einstein, Miguel de Cervantes, Ralph Waldo Emerson, Bob Dylan, etc) and most conclude with a mini- biography of a mathematician whose work relates to the chapter's content. This is an interesting book that is a pleasure to read.

According to the Preface, changes made for the tenth edition include:

1. Approximately 200 new exercises 2. Many new examples 3. New quotes 4. A freshening of the discussion portions"

- Benjamin Linowitz, Oberlin College, Published in MAA

Notations xiii
Preface xvii
0 Preliminaries
1(26)
Properties of Integers
1(4)
Modular Arithmetic
5(5)
Complex Numbers
10(2)
Mathematical Induction
12(3)
Equivalence Relations
15(3)
Functions (Mappings)
18(3)
Exercises
21(6)
1 Introduction To Groups
27(12)
Symmetries of a Square
27(3)
The Dihedral Groups
30(8)
Bibliography of Niels Abel
38(1)
2 Groups
39(20)
Definition and Examples of Groups
39(8)
Elementary Properties of Groups
47(3)
Historical Note
50(2)
Exercises
52(7)
3 Finite Groups; Subgroups
59(18)
Terminology and Notation
59(2)
Subgroup Tests
61(3)
Examples of Subgroups
64(5)
Exercises
69(8)
4 Cyclic Groups
77(20)
Properties of Cyclic Groups
77(6)
Classification of Subgroups of Cyclic Groups
83(13)
Bibliography of James Joseph Sylvester
96(1)
5 Permutation Groups
97(30)
Definitions and Notation
97(3)
Cycle Notation
100(3)
Properties of Permutations
103(12)
A Check-Digit Scheme Based on D5
115(10)
Bibliography of Augustin Cauchy
125(1)
Bibliography of Alan Taring
126(1)
6 Isomorphisms
127(22)
Motivation
127(1)
Definition and Examples
127(4)
Properties of Isomorphisms
131(3)
Automorphisms
134(4)
Cayley's Theorem
138(3)
Exercises
141(6)
Bibliography of Arthur Cayley
147(2)
7 Cosets And Lagrange's Theorem
149(22)
Properties of Cosets
149(4)
Lagrange's Theorem and Consequences
153(5)
An Application of Cosets to Permutation Groups
158(2)
The Rotation Group of a Cube and a Soccer Ball
160(3)
An Application of Cosets to the Rubik's Cube
163(1)
Exercises
163(7)
Bibliography of Joseph Lagrange
170(1)
8 External Direct Products
171(22)
Definition and Examples
171(2)
Properties of External Direct Products
173(3)
The Group of Units Modulo n as an External Direct Product
176(2)
Applications
178(6)
Exercises
184(7)
Bibliography of Leonard Adleman
191(2)
9 Normal Subgroups And Factor Groups
193(26)
Normal Subgroups
193(3)
Factor Groups
196(4)
Applications of Factor Groups
200(3)
Internal Direct Products
203(6)
Exercises
209
Bibliography of Evariste Galois
204(15)
10 Group Homomorphisms
219(22)
Definition and Examples
219(2)
Properties of Homomorphisms
221(4)
The First Isomorphism Theorem
225(7)
Exercises
232(7)
Bibliography of Camille Jordan
239(2)
11 Fundamental Theorem Of Finite Abelian Groups
241(14)
The Fundamental Theorem
241(1)
The Isomorphism Classes of Abelian Groups
242(4)
Proof of the Fundamental Theorem
246(3)
Exercises
249(6)
12 Introduction To Rings
255(12)
Motivation and Definition
255(1)
Examples of Rings
256(1)
Properties of Rings
257(2)
Subrings
259(2)
Exercises
261(5)
Bibliography of I. N. Herstein
266(1)
13 Integral Domains
267(12)
Definition and Examples
267(1)
Fields
268(3)
Characteristic of a Ring
271(2)
Exercises
273(6)
14 Ideals And Factor Rings
279(16)
Ideals
279(1)
Factor Rings
280(4)
Prime Ideals and Maximal Ideals
284(2)
Exercises
286(7)
Bibliography of Richard Dedekind
293(1)
Bibliography of Emmy Noether
294(1)
15 Ring Homomorphisms
295(16)
Definition and Examples
295(3)
Properties of Ring Homomorphisms
298(3)
The Field of Quotients
301(2)
Exercises
303(8)
16 Polynomial Rings
311(14)
Notation and Terminology
311(3)
The Division Algorithm and Consequences
314(5)
Exercises
319(6)
17 Factorization Of Polynomials
325(18)
Reducibility Tests
325(3)
Irreducibility Tests
328(6)
Unique Factorization in Z[ x]
334(1)
Weird Dice: An Application of Unique Factorization
335(3)
Exercises
338(4)
Bibliography of Serge Lang
342(1)
18 Divisibility In Integral Domains
343(22)
Irreducibles, Primes
343(3)
Historical Discussion of Fermat's Last Theorem
346(4)
Unique Factorization Domains
350(3)
Euclidean Domains
353(3)
Exercises
356(5)
Bibliography of Sophie Germain
361(1)
Bibliography of Andrew Wiles
362(1)
Bibliography of Pierre de Fermat
363(2)
19 Extension Fields
365(20)
The Fundamental Theorem of Field Theory
365(2)
Splitting Fields
367(7)
Zeros of an Irreducible Polynomial
374(5)
Exercises
379(4)
Bibliography of Leopold Kronecker
383(2)
20 Algebraic Extensions
385(16)
Characterization of Extensions
385(2)
Finite Extensions
387(5)
Properties of Algebraic Extensions
392(2)
Exercises
394(6)
Bibliography of Ernst Steinitz
400(1)
21 Finite Fields
401(18)
Classification of Finite Fields
401(1)
Structure of Finite Fields
402(6)
Subfields of a Finite Field
408(2)
Exercises
410(6)
Bibliography of L. E. Dickson
416(1)
Bibliography of E. H. Moore
417(2)
22 Geometric Constructions
419(8)
Historical Discussion of Geometric Constructions
419(1)
Constructible Numbers
420(2)
Angle-Trisectors and Circle-Squarers
422(1)
Exercises
423(4)
23 Sylow Theorems
427(20)
Conjugacy Classes
427(1)
The Class Equation
428(2)
The Sylow Theorems
430(6)
Applications of Sylow Theorems
436(4)
Exercises
440(5)
Bibliography of Ludwig Sylow
445(2)
24 Finite Simple Groups
447(20)
Historical Background
447(6)
Nonsimplicity Tests
453(5)
The Simplicity of A5
458(2)
The Fields Medal
460(1)
The Cole Prize
461(1)
Exercises
461(3)
Bibliography of Michael Aschbacher
464(1)
Bibliography of Daniel Gorenstein
465(1)
Bibliography of John Thompson
466(1)
25 Generators And Relations
467(16)
Motivation
467(1)
Definitions and Notation
468(1)
Free Group
469(2)
Generators and Relations
471(4)
Classification of Groups of Order Up to 15
475(1)
Characterization of Dihedral Groups
476(3)
Exercises
479(3)
Bibliography of Marshall Hall, Jr.
482(1)
26 Symmetry Groups
483(8)
Isometries
483(2)
Classification of Finite Plane Symmetry Groups
485(2)
Classification of Finite Groups of Rotations in R3
487(1)
Exercises
488(3)
27 Symmetry And Counting
491(12)
Motivation
491(1)
Burnside's Theorem
492(2)
Applications
494(4)
Group Action
498(1)
Exercises
499(2)
Bibliography of William Burnside
501(2)
28 Cayley Digraphs Of Groups
503(22)
Motivation
503(1)
The Cayley Digraph of a Group
503(4)
Hamiltonian Circuits and Paths
507(7)
Some Applications
514(4)
Exercises
518(4)
Bibliography of William Rowan Hamilton
522(1)
Bibliography of Paul Erdos
523(2)
29 Introduction To Algebraic Coding Theory
525(30)
Motivation
525(5)
Linear Codes
530(6)
Parity-Check Matrix Decoding
536(3)
Coset Decoding
539(4)
Historical Note
543(3)
Exercises
546(5)
Bibliography of Richard W. Hamming
551(1)
Bibliography of Jessie Mac Williams
552(1)
Bibliography of Vera Pless
553(2)
30 An Introduction To Galois Theory
555(18)
Fundamental Theorem of Galois Theory
555(7)
Solvability of Polynomials by Radicals
562(6)
Insolvability of a Quintic
568(1)
Exercises
569(4)
31 Cyclotomic Extensions
573(12)
Motivation
573(1)
Cyclotomic Polynomials
574(5)
The Constructible Regular n-gons
579(2)
Exercises
581(2)
Bibliography of Carl Friedrich Gauss
583(1)
Bibliography of Manjul Bhargava
584(1)
Selected Answers 585(44)
Index 629
Joseph A. Gallian earned his PhD from Notre Dame. In addition to receiving numerous national awards for his teaching and exposition, he has served terms as the Second Vice President, and the President of the MAA. He has served on 40 national committees, chairing ten of them. He has published over 100 articles and authored six books. Numerous articles about his work have appeared in the national news outlets, including the New York Times, the Washington Post, the Boston Globe, and Newsweek, among many others.