Algebra (Classic Version), 2nd edition

Michael Artin Massachussets Institute of Technology

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For 1- or 2-semester Algebra courses.

A modern classic

Algebra, 2nd Edition by Michael Artin is ideal for the honors undergraduate or introductory graduate course. This revision of the classic text incorporates 20 years of feedback and the author's own teaching experience. It discusses concrete topics of algebra in greater detail than most texts, preparing students for the more abstract concepts; linear algebra is tightly integrated throughout.

This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price.

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.

New and updated features of this title

Exercises have been added throughout the book.
Extensive rewriting incorporates 20 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 the representations of SU2 is improved (Chapter 10).
The discussion of function fields has been improved (Chapter 15). 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, including a short section on using continuity to deduce facts about linear operators (Chapter 5); 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).

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

About our author

Michael Artin received his A.B. from Princeton University in 1955 and his M.A. and Ph.D. from Harvard University in 1956 and 1960, respectively. He continued at Harvard as Benjamin Peirce Lecturer, 1960 - 63. He joined the MIT mathematics faculty in 1963, and was appointed Norbert Wiener Professor from 1988 - 93. Outside MIT, Artin served as President of the American Mathematical Society from 1990-92. He has received honorary doctorate degrees from the University of Antwerp and University of Hamburg.

Professor Artin is an algebraic geometer, concentrating on non-commutative algebra. He has received many awards throughout his distinguished career, including the Undergraduate Teaching Prize and the Educational and Graduate Advising Award. He received the Leroy P. Steele Prize for Lifetime Achievement from the AMS. In 2005 he was honored with the Harvard Graduate School of Arts & Sciences Centennial Medal, for being "an architect of the modern approach to algebraic geometry." Professor Artin is a Member of the National Academy of Sciences, Fellow of the American Academy of Arts & Sciences, Fellow of the American Association for the Advancement of Science, and Fellow of the Society of Industrial and Applied Mathematics. He is a Foreign Member of the Royal Holland Society of Sciences, and Honorary Member of the Moscow Mathematical Society.

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