1. Trang chủ
  2. » Khoa Học Tự Nhiên

Joseph gallian contemporary abstract algebra

646 4 0

Đang tải... (xem toàn văn)

Tài liệu hạn chế xem trước, để xem đầy đủ mời bạn chọn Tải xuống

THÔNG TIN TÀI LIỆU

Cấu trúc

  • Front Cover

  • Title Page

  • Copyright

  • Contents

  • Preface

  • PART 1: Integers and Equivalence Relations

    • 0 Preliminaries

      • Properties of Integers

      • Modular Arithmetic

      • Mathematical Induction

      • Equivalence Relations

      • Functions (Mappings)

      • Exercises

      • Computer Exercises

  • PART 2: Groups

    • 1 Introduction to Groups

      • Symmetries of a Square

      • The Dihedral Groups

      • Exercises

      • Biography of Niels Abel

    • 2 Groups

      • Definition and Examples of Groups

      • Elementary Properties of Groups

      • Historical Note

      • Exercises

      • Computer Exercises

    • 3 Finite Groups; Subgroups

      • Terminology and Notation

      • Subgroup Tests

      • Examples of Subgroups

      • Exercises

      • Computer Exercises

    • 4 Cyclic Groups

      • Properties of Cyclic Groups

      • Classification of Subgroups of Cyclic Groups

      • Exercises

      • Computer Exercises

      • Biography of J. J. Sylvester

      • Supplementary Exercises for Chapters 1–4

    • 5 Permutation Groups

      • Definition and Notation

      • Cycle Notation

      • Properties of Permutations

      • Exercises

      • Computer Exercises

      • Biography of Augustin Cauchy

    • 6 Isomorphisms

      • Motivation

      • Definition and Examples

      • Cayley’s Theorem

      • Automorphisms

      • Exercises

      • Computer Exercise

      • 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

      • Exercises

      • Computer Exercise

      • 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

      • Computer Exercises

      • Biography of Leonard Adleman

      • Supplementary Exercises for Chapters 5–8

    • 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

      • Computer Exercise

      • 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

      • Computer Exercises

      • Supplementary Exercises for Chapters 9–11

  • PART 3: Rings

    • 12 Introduction to Rings 237

      • Motivation and Definition

      • Examples of Rings

      • Properties of Rings

      • Subrings

      • Exercises

      • Computer Exercises

      • Biography of I. N. Herstein

    • 13 Integral Domains

      • Definition and Examples

      • Fields

      • Characteristic of a Ring

      • Exercises

      • Computer Exercises

      • Biography of Nathan Jacobson

    • 14 Ideals and Factor Rings

      • Ideals

      • Factor Rings

      • Prime Ideals and Maximal Ideals

      • Exercises

      • Computer Exercises

      • Biography of Richard Dedekind

      • Biography of Emmy Noether

      • Supplementary Exercises for Chapters 12–14

    • 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

      • Biography of Saunders Mac Lane

    • 17 Factorization of Polynomials

      • Reducibility Tests

      • Irreducibility Tests

      • Unique Factorization in Z[x]

      • Factorization

      • Exercises

      • Computer 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

      • Computer Exercise

      • Biography of Sophie Germain

      • Biography of Andrew Wiles

      • Supplementary Exercises for Chapters 15–18

  • PART 4: Fields

    • 19 Vector Spaces

      • Definition and Examples

      • Subspaces

      • Linear Independence

      • Exercises

      • Biography of Emil Artin

      • Biography of Olga Taussky-Todd

    • 20 Extension Fields

      • The Fundamental Theorem of Field Theory

      • Splitting Fields

      • Zeros of an Irreducible Polynomial

      • Exercises

      • Biography of Leopold Kronecker

    • 21 Algebraic Extensions

      • Characterization of Extensions

      • Finite Extensions

      • Properties of Algebraic Extensions

      • Exercises

      • Biography of Irving Kaplansky

    • 22 Finite Fields

      • Classification of Finite Fields

      • Structure of Finite Fields

      • Subfields of a Finite Field

      • Exercises

      • Computer Exercises

      • Biography of L. E. Dickson

    • 23 Geometric Constructions

      • Historical Discussion of Geometric Constructions

      • Constructible Numbers

      • Angle-Trisectors and Circle-Squarers

      • Exercises

      • Supplementary Exercises for Chapters 19–23

  • PART 5: Special Topics

    • 24 Sylow Theorems

      • Conjugacy Classes

      • The Class Equation

      • The Probability That Two Elements Commute

      • The Sylow Theorems

      • Applications of Sylow Theorems

      • Computer Exercise

      • Biography of Ludwig Sylow

    • 25 Finite Simple Groups

      • Historical Background

      • Nonsimplicity Tests

      • The Simplicity of A5

      • The Fields Medal

      • The Cole Prize

      • Exercises

      • Computer Exercises

      • Biography of Michael Aschbacher

      • Biography of Daniel Gorenstein

      • Biography of John Thompson

    • 26 Generators and Relations

      • Motivation

      • Definitions and Notation

      • Free Group

      • Generators and Relations

      • Classification of Groups of Order Up to 15

      • Characterization of Dihedral Groups

      • Realizing the Dihedral Groups with Mirrors

      • Exercises

      • Biography of Marshall Hall, Jr.

    • 27 Symmetry Groups

      • Isometries

      • Classification of Finite Plane Symmetry Groups

      • Classification of Finite Groups of Rotations in R3

      • Exercises

    • 28 Frieze Groups and Crystallographic Groups

      • The Frieze Groups

      • The Crystallographic Groups

      • Identification of Plane Periodic Patterns

      • Exercises

      • Biography of M. C. Escher

      • Biography of George Pólya

      • Biography of John H. Conway

    • 29 Symmetry and Counting

      • Motivation

      • Burnside’s Theorem

      • Applications

      • Group Action

      • Exercises

      • Biography of William Burnside

    • 30 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 Erdös

    • 31 Introduction to Algebraic Coding Theory

      • Motivation

      • Linear Codes

      • Parity-Check Matrix Decoding

      • Coset Decoding

      • Historical Note: The Ubiquitous Reed-Solomon Codes

      • Exercises

      • Biography of Richard W. Hamming

      • Biography of Jessie MacWilliams

      • Biography of Vera Pless

    • 32 An Introduction to Galois Theory

      • Fundamental Theorem of Galois Theory

      • Solvability of Polynomials by Radicals

      • Insolvability of a Quintic

      • Exercises

      • Biography of Philip Hall

    • 33 Cyclotomic Extensions

      • Motivation

      • Cyclotomic Polynomials

      • The Constructible Regular n-gons

      • Exercises

      • Computer Exercise

      • Biography of Carl Friedrich Gauss

      • Biography of Manjul Bhargava

      • Supplementary Exercises for Chapters 24–33

  • Selected Answers

  • Text Credits

  • Photo Credits

  • Index of Mathematicians

  • Index of Terms

Nội dung

Ngày đăng: 27/05/2022, 10:18

TÀI LIỆU CÙNG NGƯỜI DÙNG

TÀI LIỆU LIÊN QUAN