Michael Artin  
Total pages
November 2013
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Algebra, Second Edition, by Michael Artin, is ideal for the honors undergraduate or introductory graduate course. The second edition of this classic text incorporates twenty years of feedback and the author’s own teaching experience. The text discusses concrete topics of algebra in greater detail than most texts, preparing students for the more abstract concepts; linear algebra is tightly integrated throughout.


  • High emphasis on concrete topics, such as symmetry, linear groups, quadratic number fields, and lattices, prepares students to learn more abstract concepts. The focus on these special topics also allows some abstractions to be treated more concisely, devoting more space to the areas students are the most interested in.
  • Thechapter 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 from others. 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.

New to this Edition

  • Exercises have been added throughout the book.
  • This text has been rewritten extensively, incorporating twenty years’ worth of user feedback and the author’s own teaching experience.
    • Permutations are introduced early, and computation with them is clarified.
    • Correspondence Theorem discussion (Chapter 2) has been revised
    • Linear Transformations coverage has been split into two (now Chapters 4 and 5)
    • Jordan Form is presented earlier, with Fillipov's proof (now in Chapter 4).
    • The proof of the orthogonality relations for characters has been improved (Chapter 10).
    • The discussion of function fields has been improved (Chapter 15).
    • The discussion of the representations of SU2 is improved (Chapter 10).
    • The chapter on Factorization has been split into two (now Chapters 12 and 13), with quadratic number fields appearing in the latter chapter.
  • New topics have been introduced throughout the text.
    • A short section on using continuity to deduce facts about linear operators (Chapter 5)
    • A proof that the alternating groups are simple (Chapter 7)
    • A section on spheres (Chapter 9)
    • A section on product rings (Chapter 11)
    • A short discussion of computer methods to factor polynomials (Chapter 12)
    • Cauchy's theorem bounding the roots of a polynomial (Chapter 12)
    • A proof of the splitting theorem based on symmetric functions (Chapter 16)


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 GroupSU2

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 v(-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