Algebra

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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

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