Michael Artin  
Total pages
November 2013
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Algebra, 2nd Edition, by Michael Artin, is ideal for the honors undergraduate or introductory graduate course. This 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.
  • The chapter organisation emphasises 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
  • 2. Groups
  • 3. Vector Spaces
  • 4. Linear Operators
  • 5. Applications of Linear Operators
  • 6. Symmetry
  • 7. More Group Theory
  • 8. Bilinear Forms
  • 9. Linear Groups
  • 10. Group Representations
  • 11. Rings
  • 12. Factoring
  • 13. Quadratic Number Fields
  • 14. Linear Algebra in a Ring
  • 15. Fields