Contemporary Abstract Algebra
Námskeið
- STÆ303G Algebra I.
Ensk lýsing:
CONTEMPORARY ABSTRACT ALGEBRA, NINTH EDITION provides a solid introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working mathematicians, computer scientists, physicists, and chemists. The text includes numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings giving the subject a current feel which makes the content interesting and relevant for students.
Lýsing:
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 Instructor’s Solutions Manual written by the author has comprehensive solutions for all exercises and some alternative solutions to develop a critical thought and deeper understanding.
Annað
- Höfundur: Joseph A. Gallian
- Útgáfa:10
- Útgáfudagur: 2021-01-19
- Hægt að prenta út 2 bls.
- Hægt að afrita 2 bls.
- Format:ePub
- ISBN 13: 9781000337358
- Print ISBN: 9780367651787
- ISBN 10: 1000337359
Efnisyfirlit
- Cover Page
- Half-Title Page
- Series Page
- Title Page
- Copyright Page
- Dedication Page
- Contents
- Notations
- Preface
- 0 Preliminaries
- Properties of Integers
- Modular Arithmetic
- Complex Numbers
- Mathematical Induction
- Equivalence Relations
- Functions (Mappings)
- Exercises
- 1 Introduction to Groups
- Symmetries of a Square
- The Dihedral Groups
- Biography of Niels Abel
- 2 Groups
- Definition and Examples of Groups
- Elementary Properties of Groups
- Historical Note
- Exercises
- 3 Finite Groups; Subgroups
- Terminology and Notation
- Subgroup Tests
- Examples of Subgroups
- Exercises
- 4 Cyclic Groups
- Properties of Cyclic Groups
- Classification of Subgroups of Cyclic Groups
- Biography of James Joseph Sylvester
- 5 Permutation Groups
- Definitions and Notation
- Cycle Notation
- Properties of Permutations
- A Check-Digit Scheme Based on D_5
- Biography of Augustin Cauchy
- Biography of Alan Turing
- 6 Isomorphisms
- Motivation
- Definition and Examples
- Properties of Isomorphisms
- Automorphisms
- Cayley’s Theorem
- Exercises
- Biography of Arthur Cayley
- 7 Cosets and Lagrange's Theorem
- Properties of Cosets
- Lagrange's Theorem and Consequences
- An Application of Cosets to Permutation Groups
- The Rotation Group of a Cube and a Soccer Ball
- An Application of Cosets to the Rubik's Cube
- Exercises
- Biography of Joseph Lagrange
- 8 External Direct Products
- Definition and Examples
- Properties of External Direct Products
- The Group of Units Modulo n as an External Direct Product
- Applications
- Exercises
- Biography of Leonard Adleman
- 9 Normal Subgroups and Factor Groups
- Normal Subgroups
- Factor Groups
- Applications of Factor Groups
- Internal Direct Products
- Exercises
- Biography of Évariste Galois
- 10 Group Homomorphisms
- Definition and Examples
- Properties of Homomorphisms
- The First Isomorphism Theorem
- Exercises
- Biography of Camille Jordan
- 11 Fundamental Theorem of Finite Abelian Groups
- The Fundamental Theorem
- The Isomorphism Classes of Abelian Groups
- Proof of the Fundamental Theorem
- Exercises
- 12 Introduction to Rings
- Motivation and Definition
- Examples of Rings
- Properties of Rings
- Subrings
- Exercises
- Biography of I. N. Herstein
- 13 Integral Domains
- Definition and Examples
- Fields
- Characteristic of a Ring
- Exercises
- 14 Ideals and Factor Rings
- Ideals
- Factor Rings
- Prime Ideals and Maximal Ideals
- Exercises
- Biography of Richard Dedekind
- Biography of Emmy Noether
- 15 Ring Homomorphisms
- Definition and Examples
- Properties of Ring Homomorphisms
- The Field of Quotients
- Exercises
- 16 Polynomial Rings
- Notation and Terminology
- The Division Algorithm and Consequences
- Exercises
- 17 Factorization of Polynomials
- Reducibility Tests
- Irreducibility Tests
- Unique Factorization in Z[x]
- Weird Dice: An Application of Unique Factorization
- Exercises
- Biography of Serge Lang
- 18 Divisibility in Integral Domains
- Irreducibles, Primes
- Historical Discussion of Fermat's Last Theorem
- Unique Factorization Domains
- Euclidean Domains
- Exercises
- Biography of Sophie Germain
- Biography of Andrew Wiles
- Biography of Pierre de Fermat
- 19 Extension Fields
- The Fundamental Theorem of Field Theory
- Splitting Fields
- Zeros of an Irreducible Polynomial
- Exercises
- Biography of Leopold Kronecker
- 20 Algebraic Extensions
- Characterization of Extensions
- Finite Extensions
- Properties of Algebraic Extensions
- Exercises
- Biography of Ernst Steinitz
- 21 Finite Fields
- Classification of Finite Fields
- Structure of Finite Fields
- Subfields of a Finite Field
- Exercises
- Biography of L. E. Dickson
- Biography of E. H. Moore
- 22 Geometric Constructions
- Historical Discussion of Geometric Constructions
- Constructible Numbers
- Angle-Trisectors and Circle-Squarers
- Exercises
- 23 Sylow Theorems
- Conjugacy Classes
- The Class Equation
- The Sylow Theorems
- Applications of Sylow Theorems
- Exercises
- Biography of Ludwig Sylow
- 24 Finite Simple Groups
- Historical Background
- Nonsimplicity Tests
- The Simplicity of A_5
- The Fields Medal
- The Cole Prize
- Exercises
- Biography of Michael Aschbacher
- Biography of Daniel Gorenstein
- Biography of John Thompson
- 25 Generators and Relations
- Motivation
- Definitions and Notation
- Free Group
- Generators and Relations
- Classification of Groups of Order Up to 15
- Characterization of Dihedral Groups
- Exercises
- Biography of Marshall Hall, Jr.
- 26 Symmetry Groups
- Isometries
- Classification of Finite Plane Symmetry Groups
- Classification of Finite Groups of Rotations in R^3
- Exercises
- 27 Symmetry and Counting
- Motivation
- Burnside's Theorem
- Applications
- Group Action
- Exercises
- Biography of William Burnside
- 28 Cayley Digraphs of Groups
- Motivation
- The Cayley Digraph of a Group
- Hamiltonian Circuits and Paths
- Some Applications
- Exercises
- Biography of William Rowan Hamilton
- Biography of Paul Erdős
- 29 Introduction to Algebraic Coding Theory
- Motivation
- Linear Codes
- Parity-Check Matrix Decoding
- Coset Decoding
- Historical Note
- Exercises
- Biography of Richard W. Hamming
- Biography of Jessie MacWilliams
- Biography of Vera Pless
- 30 An Introduction to Galois Theory
- Fundamental Theorem of Galois Theory
- Solvability of Polynomials by Radicals
- Insolvability of a Quintic
- Exercises
- 31 Cyclotomic Extensions
- Motivation
- Cyclotomic Polynomials
- The Constructible Regular n-gons
- Exercises
- Biography of Carl Friedrich Gauss
- Biography of Manjul Bhargava
- Selected Answers
- Index
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- Gerð : 208
- Höfundur : 16997
- Útgáfuár : 2021
- Leyfi : 380
