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Algebra (Classic Version), 2nd edition

  • Michael Artin

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Overview

Hallmark features of this title

  • High emphasis on concrete topics such as symmetry, linear groups, quadratic number fields, and lattices prepares students to learn more abstract concepts. This focus also allows some abstractions to be treated more concisely.
  • The chapter organization emphasizes the connections between algebra and geometry at the start, with the beginning chapters containing the content most important for students in other fields. To counter the fact that arithmetic receives less initial emphasis, the later chapters have a strong arithmetic slant.
  • Treatment beyond the basics sets this book apart. Students with a reasonably mature mathematical background will benefit from the relatively informal treatments the author gives to the more advanced topics.
  • Content notes in the preface include teaching tips from the author’s own classroom experience.
  • Challenging exercises are indicated with an asterisk, allowing instructors to easily create the right assignments for their class.

Published by Pearson (June 1st 2023) - Copyright © 2023

ISBN-13: 9780137980994

Subject: Advanced Math

Category: Abstract Algebra

Table of contents

1. Matrices

  • 1.1 The Basic Operations
  • 1.2 Row Reduction
  • 1.3 The Matrix Transpose
  • 1.4 Determinants
  • 1.5 Permutations
  • 1.6 Other Formulas for the Determinant
  • 1.7 Exercises

2. Groups

  • 2.1 Laws of Composition
  • 2.2 Groups and Subgroups
  • 2.3 Subgroups of the Additive Group of Integers
  • 2.4 Cyclic Groups
  • 2.5 Homomorphisms
  • 2.6 Isomorphisms
  • 2.7 Equivalence Relations and Partitions
  • 2.8 Cosets
  • 2.9 Modular Arithmetic
  • 2.10 The Correspondence Theorem
  • 2.11 Product Groups
  • 2.12 Quotient Groups
  • 2.13 Exercises

3. Vector Spaces

  • 3.1 Subspaces of Rn
  • 3.2 Fields
  • 3.3 Vector Spaces
  • 3.4 Bases and Dimension
  • 3.5 Computing with Bases
  • 3.6 Direct Sums
  • 3.7 Infinite-Dimensional Spaces
  • 3.8 Exercises

4. Linear Operators

  • 4.1 The Dimension Formula
  • 4.2 The Matrix of a Linear Transformation
  • 4.3 Linear Operators
  • 4.4 Eigenvectors
  • 4.5 The Characteristic Polynomial
  • 4.6 Triangular and Diagonal Forms
  • 4.7 Jordan Form
  • 4.8 Exercises

5. Applications of Linear Operators

  • 5.1 Orthogonal Matrices and Rotations
  • 5.2 Using Continuity
  • 5.3 Systems of Differential Equations
  • 5.4 The Matrix Exponential
  • 5.5 Exercises

6. Symmetry

  • 6.1 Symmetry of Plane Figures
  • 6.2 Isometries
  • 6.3 Isometries of the Plane
  • 6.4 Finite Groups of Orthogonal Operators on the Plane
  • 6.5 Discrete Groups of Isometries
  • 6.6 Plane Crystallographic Groups
  • 6.7 Abstract Symmetry: Group Operations
  • 6.8 The Operation on Cosets
  • 6.9 The Counting Formula
  • 6.10 Operations on Subsets
  • 6.11 Permutation Representation
  • 6.12 Finite Subgroups of the Rotation Group
  • 6.13 Exercises

7. More Group Theory

  • 7.1 Cayley's Theorem
  • 7.2 The Class Equation
  • 7.3 r-groups
  • 7.4 The Class Equation of the Icosahedral Group
  • 7.5 Conjugation in the Symmetric Group
  • 7.6 Normalizers
  • 7.7 The Sylow Theorems
  • 7.8 Groups of Order 12
  • 7.9 The Free Group
  • 7.10 Generators and Relations
  • 7.11 The Todd-Coxeter Algorithm
  • 7.12 Exercises

8. Bilinear Forms

  • 8.1 Bilinear Forms
  • 8.2 Symmetric Forms
  • 8.3 Hermitian Forms
  • 8.4 Orthogonality
  • 8.5 Euclidean spaces and Hermitian spaces
  • 8.6 The Spectral Theorem
  • 8.7 Conics and Quadrics
  • 8.8 Skew-Symmetric Forms
  • 8.9 Summary
  • 8.10 Exercises

9. Linear Groups

  • 9.1 The Classical Groups
  • 9.2 Interlude: Spheres
  • 9.3 The Special Unitary Group SU2
  • 9.4 The Rotation Group SO3
  • 9.5 One-Parameter Groups
  • 9.6 The Lie Algebra
  • 9.7 Translation in a Group
  • 9.8 Normal Subgroups of SL2
  • 9.9 Exercises

10. Group Representations

  • 10.1 Definitions
  • 10.2 Irreducible Representations
  • 10.3 Unitary Representations
  • 10.4 Characters
  • 10.5 One-Dimensional Characters
  • 10.6 The Regular Representations
  • 10.7 Schur's Lemma
  • 10.8 Proof of the Orthogonality Relations
  • 10.9 Representationsof SU2
  • 10.10 Exercises

11. Rings

  • 11.1 Definition of a Ring
  • 11.2 Polynomial Rings
  • 11.3 Homomorphisms and Ideals
  • 11.4 Quotient Rings
  • 11.5 Adjoining Elements
  • 11.6 Product Rings
  • 11.7 Fraction Fields
  • 11.8 Maximal Ideals
  • 11.9 Algebraic Geometry
  • 11.10 Exercises

12. Factoring

  • 12.1 Factoring Integers
  • 12.2 Unique Factorization Domains
  • 12.3 Gauss's Lemma
  • 12.4 Factoring Integer Polynomial
  • 12.5 Gauss Primes
  • 12.6 Exercises

13. Quadratic Number Fields

  • 13.1 Algebraic Integers
  • 13.2 Factoring Algebraic Integers
  • 13.3 Ideals in Z √(-5)
  • 13.4 Ideal Multiplication
  • 13.5 Factoring Ideals
  • 13.6 Prime Ideals and Prime Integers
  • 13.7 Ideal Classes
  • 13.8 Computing the Class Group
  • 13.9 Real Quadratic Fields
  • 13.10 About Lattices
  • 13.11 Exercises

14. Linear Algebra in a Ring

  • 14.1 Modules
  • 14.2 Free Modules
  • 14.3 Identities
  • 14.4 Diagonalizing Integer Matrices
  • 14.5 Generators and Relations
  • 14.6 Noetherian Rings
  • 14.7 Structure to Abelian Groups
  • 14.8 Application to Linear Operators
  • 14.9 Polynomial Rings in Several Variables
  • 14.10 Exercises

15. Fields

  • 15.1 Examples of Fields
  • 15.2 Algebraic and Transcendental Elements
  • 15.3 The Degree of a Field Extension
  • 15.4 Finding the Irreducible Polynomial
  • 15.5 Ruler and Compass Constructions
  • 15.6 Adjoining Roots
  • 15.7 Finite Fields
  • 15.8 Primitive Elements
  • 15.9 Function Fields
  • 15.10 The Fundamental Theorem of Algebra
  • 15.11 Exercises

16. Galois Theory

  • 16.1 Symmetric Functions
  • 16.2 The Discriminant
  • 16.3 Splitting Fields
  • 16.4 Isomorphisms of Field Extensions
  • 16.5 Fixed Fields
  • 16.6 Galois Extensions
  • 16.7 The Main Theorem
  • 16.8 Cubic Equations
  • 16.9 Quartic Equations
  • 16.10 Roots of Unity
  • 16.11 Kummer Extensions
  • 16.12 Quintic Equations
  • 16.13 Exercises

Appendix A. Background Material

  • A.1 About Proofs
  • A.2 The Integers
  • A.3 Zorn's Lemma
  • A.4 The Implicit Function Theorem
  • A.5 Exercises

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