Algebra, Second Edition

Algebra, Second Edition , by Michael Artin, provides comprehensive coverage at the level of an honors-undergraduate or introductory-graduate course. The second edition of this classic text incorporates twenty years of feedback plus the author’s own teaching experience. This book discusses concrete topics of algebra in greater detail than others, preparing readers for the more abstract concepts; linear algebra is tightly integrated throughout. Contents Preface 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 Exercise 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 Exercis 3 Vector Spaces 3.1 Subspaces of R^n 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 Exercise 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 Exercise 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 Exercise 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 Representations 6.12 Finite Subgroups of the Rotation Group Exercise 7 More Group Theory 7.1 Cayley’s Theorem 7.2 The Class Equation 7.3 p-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 Exercises 8 Bilinear Forms 8.1 Bilinear Forms 8.2 Symmetric Forms 8.3 Hermitian Form 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 Exercises 9 Linear Groups 9.1 The Classical Groups 9.2 Interlude: Sphere 9.3 The Special Unitary Group SU2 9.4 The Rotation Group SO3 9.5 One-Parameter Groups 9.6 The Lie Algebr 9.7 Translation in a Group 9.8 Normal Subgroups of SL2 Exercise 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 Representation 10.7 Schur’s Lemma 10.8 Proof of the Orthogonality Relations 10.9 Representations of SU2 Exercises 11 Rings 11.1 Definition of a R in 11.2 Polynomial Rings 11.3 Homomorphisms and Ideals 11.4 Quotient Rings 11.5 Adjoining Elements 11.6 Product Rings 11.7 Fractions 11.8 Maximal Ideals 11.9 Algebraic Geometry Exercises 12 Factoring 12.1 Factoring Integers 12.2 Unique Factorization Domains 12.3 Gauss’s Lemma 12.4 Factoring Integer Polynomials 12.5 Gauss Primes Exercise 13 Quadratic Number Fields 13.1 Algebraic Integers 13.2 Factoring Algebraic Integers 13.3 Ideals in Z[(-5)^(-1/2)] 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 Exercises 14 Linear Algebra in a Ring 14.1 Modules 14.2 Free Modules 14.3 Identitie 14.4 Diagonalizing Integer Matrices 14.5 Generators and Relations 14.6 Noetherian Rings 14.7 Structure of Abelian Groups 14.8 Application to Linear Operators 14.9 Polynomial Rings in Several Variables 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 Exercise 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 Exercises APPENDIX Background Material A.1 About Proofs A.2 The Integers A.3 Zorn’s Lemma A.4 The Implicit Function Theorem Exercises Bibliography Notation Index 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. -- Publisher's Description. Matrices -- Groups -- Vector Spaces -- Linear Operators -- Applications Of Linear Operators -- Symmetry -- More Group Theory -- Bilinear Forms -- Linear Groups -- Group Representations -- Rings -- Factoring -- Quadratic Number Fields -- Linear Algebra In A Ring -- Fields -- Galois Theory. Michael Artin. Includes Bibliographical References (p. 523-524) And Index.