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Full Description
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
1. Matrices1.1 The Basic Operations1.2 Row Reduction1.3 The Matrix Transpose1.4 Determinants1.5 Permutations1.6 Other Formulas for the Determinant1.7 Exercises2. Groups2.1 Laws of Composition2.2 Groups and Subgroups2.3 Subgroups of the Additive Group of Integers2.4 Cyclic Groups2.5 Homomorphisms2.6 Isomorphisms2.7 Equivalence Relations and Partitions2.8 Cosets2.9 Modular Arithmetic2.10 The Correspondence Theorem2.11 Product Groups2.12 Quotient Groups2.13 Exercises3. Vector Spaces3.1 Subspaces of Rn3.2 Fields3.3 Vector Spaces3.4 Bases and Dimension3.5 Computing with Bases3.6 Direct Sums3.7 Infinite-Dimensional Spaces3.8 Exercises4. Linear Operators4.1 The Dimension Formula4.2 The Matrix of a Linear Transformation4.3 Linear Operators4.4 Eigenvectors4.5 The Characteristic Polynomial4.6 Triangular and Diagonal Forms4.7 Jordan Form4.8 Exercises5. Applications of Linear Operators5.1 Orthogonal Matrices and Rotations5.2 Using Continuity5.3 Systems of Differential Equations5.4 The Matrix Exponential5.5 Exercises6. Symmetry6.1 Symmetry of Plane Figures6.2 Isometries6.3 Isometries of the Plane6.4 Finite Groups of Orthogonal Operators on the Plane6.5 Discrete Groups of Isometries6.6 Plane Crystallographic Groups6.7 Abstract Symmetry: Group Operations6.8 The Operation on Cosets6.9 The Counting Formula6.10 Operations on Subsets6.11 Permutation Representation6.12 Finite Subgroups of the Rotation Group6.13 Exercises7. More Group Theory7.1 Cayley's Theorem7.2 The Class Equation7.3 r-groups7.4 The Class Equation of the Icosahedral Group7.5 Conjugation in the Symmetric Group7.6 Normalizers7.7 The Sylow Theorems7.8 Groups of Order 127.9 The Free Group7.10 Generators and Relations7.11 The Todd-Coxeter Algorithm7.12 Exercises8. Bilinear Forms8.1 Bilinear Forms8.2 Symmetric Forms8.3 Hermitian Forms8.4 Orthogonality8.5 Euclidean spaces and Hermitian spaces8.6 The Spectral Theorem8.7 Conics and Quadrics8.8 Skew-Symmetric Forms8.9 Summary8.10 Exercises9. Linear Groups9.1 The Classical Groups9.2 Interlude: Spheres9.3 The Special Unitary GroupSU29.4 The Rotation Group SO39.5 One-Parameter Groups9.6 The Lie Algebra9.7 Translation in a Group9.8 Normal Subgroups of SL29.9 Exercises10. Group Representations10.1 Definitions10.2 Irreducible Representations10.3 Unitary Representations10.4 Characters10.5 One-Dimensional Characters10.6 The Regular Representations10.7 Schur's Lemma10.8 Proof of the Orthogonality Relations10.9 Representationsof SU210.10 Exercises11. Rings11.1 Definition of a Ring11.2 Polynomial Rings11.3 Homomorphisms and Ideals11.4 Quotient Rings11.5 Adjoining Elements11.6 Product Rings11.7 Fraction Fields11.8 Maximal Ideals11.9 Algebraic Geometry11.10 Exercises12. Factoring12.1 Factoring Integers12.2 Unique Factorization Domains12.3 Gauss's Lemma12.4 Factoring Integer Polynomial12.5 Gauss Primes12.6 Exercises13. Quadratic Number Fields13.1 Algebraic Integers13.2 Factoring Algebraic Integers13.3 Ideals in Z v(-5)13.4 Ideal Multiplication13.5 Factoring Ideals13.6 Prime Ideals and Prime Integers13.7 Ideal Classes13.8 Computing the Class Group13.9 Real Quadratic Fields13.10 About Lattices13.11 Exercises14. Linear Algebra in a Ring14.1 Modules14.2 Free Modules14.3 Identities14.4 Diagonalizing Integer Matrices14.5 Generators and Relations14.6 Noetherian Rings14.7 Structure to Abelian Groups14.8 Application to Linear Operators14.9 Polynomial Rings in Several Variables14.10 Exercises15. Fields15.1 Examples of Fields15.2 Algebraic and Transcendental Elements15.3 The Degree of a Field Extension15.4 Finding the Irreducible Polynomial15.5 Ruler and Compass Constructions15.6 Adjoining Roots15.7 Finite Fields15.8 Primitive Elements15.9 Function Fields15.10 The Fundamental Theorem of Algebra15.11 Exercises
