Garrett Birkhoff (1911–1996)

Indeholder "Preface to the Second Edition", "List of Symbols", "Chapter 1. Sets, Functions, and Integers", " 1. Sets", " 2. Functions", " 3. Relations and Binary Operations", " 4. The Natural Numbers", " 5. Addition and Multiplication", " 6. Inequalities", " 7. The Integers", " 8. The Integers Modulo n", " 9. Equivalence Relations and Quotient Sets", " 10. Morphisms", " 11. Semigroups and Monoids", "Chapter 2. Groups", " 1. Groups and Symmetry", " 2. Rules of Calculation", " 3. Cyclic Groups", " 4. Subgroups", " 5. Defining Relations", " 6. Symmetric and Alternating Groups", " 7. Transformation Groups", " 8. Cosets", " 9. Kernel and Image", " 10. Quotient Groups", "Chapter 3. Rings", " 1. Axioms for Rings", " 2. Constructions for Rings", " 3. Quotient Rings", " 4. Integral Domains and Fields", " 5. The Field of Quotients", " 6. Polynomials", " 7. Polynomials as Functions", " 8. The Division Algorithm", " 9. Principal Ideal Domains", " 10. Unique Factorization", " 11. Prime Fields", " 12. The Euclidean Algorithm", " 13. Commutative Quotient Rings", "Chapter 4. Universal Constructions", " 1. Examples of Universals", " 2. Functors", " 3. Universal Elements", " 4. Polynomials in Several Variables", " 5. Categories", " 6. Posets and Lattices", " 7. Contravariance and Duality", " 8. The Category of Sets", " 9. The Category of Finite Sets", "Chapter 5. Modules", " 1. Sample Modules", " 2. Linear Transformations", " 3. Submodules", " 4. Quotient Modules", " 5. Free Modules", " 6. Biproducts", " 7. Dual Modules", "Chapter 6. Vector Spaces", " 1. Bases and Coordinates", " 2. Dimension", " 3. Constructions for Bases", " 4. Dually Paired Vector Spaces", " 5. Elementary Operations", " 6. Systems of Linear Equations", "Chapter 7. Matrices", " 1. Matrices and Free Modules", " 2. Matrices and Biproducts", " 3. The Matrix of a Map", " 4. The Matrix of a Composite", " 5. Ranks of Matrices", " 6. Invertible Matrices", " 7. Change of Bases", " 8. Eigenvectors and Eigenvalues", "Chapter 8. Special Fields", " 1. Ordered Domains", " 2. The Ordered Field Q", " 3. Polynomial Equations", " 4. Convergence in Ordered Fields", " 5. The Real Field R", " 6. Polynomials over R", " 7. The Complex Plane", " 8. The Quaternions", " 9. Extended Formal Power Series", " 10. Valuations and p-adic Numbers", "Chapter 9. Determinants and Tensor Products", " 1. Multilinear and Alternating Functions", " 2. Determinants of Matrices", " 3. Cofactors and Cramer's Rule", " 4. Determinants of Maps", " 5. The Characteristic Polynomial", " 6. The Minimal Polynomial", " 7. Universal Bilinear Functions", " 8. Tensor Products", " 9. Exact Sequences", " 10. Identities on Tensor Products", " 11. Change of Rings", " 12. Algebras", "Chapter 10. Bilinear and Quadratic Forms", " 1. Bilinear Forms", " 2. Symmetric Matrices", " 3. Quadratic Forms", " 4. Real Quadratic Forms", " 5. Inner Products", " 6. Orthonormal Bases", " 7. Orthogonal Matrices", " 8. The Principal Axis Theorem", " 9. Unitary Spaces", " 10. Normal Matrices", "Chapter 11. Similar Matrices and Finite Abelian Groups", " 1. Noetherian Modules", " 2. Cyclic Modules", " 3. Torsion Modules", " 4. The Rational Canonical Form for Matrices", " 5. Primary Modules", " 6. Free Modules", " 7. Equivalence of Matrices", " 8. The Calculation of Invariant Factors", "Chapter 12. Structure of Groups", " 1. Isomorphism Theorems", " 2. Group Extensions", " 3. Characteristic Subgroups", " 4. Conjugate Classes", " 5. The Sylow Theorems", " 6. Nilpotent Groups", " 7. Solvable Groups", " 8. The Jordan-Hölder Theorem", " 9. Simplicity of A_n", "Chapter 13. Galois Theory", " 1. Quadric and Cubic Equations", " 2. Algebraic and Transcendental Elements", " 3. Degrees", " 4. Ruler and Compass", " 5. Splitting Fields", " 6. Galois Group of Polynomials", " 7. Separable Polynomials", " 8. Finite Fields", " 9. Normal Extensions", " 10. The Fundamental Theorem", " 11. The Solution of Equations by Radicals", "Chapter 14. Lattices", " 1. Posets: Duality Principle", " 2. Lattice Identities", " 3. Sublattices and Products of Lattices", " 4. Modular Lattices", " 5. Jordan-Hölder-Dedekind Theorem", " 6. Distributive Lattices", " 7. Rings of Sets", " 8. Boolean Algebras", " 9. Free Boolean Algebras", "Chapter 15. Categories and Adjoint Functors", " 1. Categories", " 2. Functors", " 3. Contravariant Functors ", " 4. Natural Transformations", " 5. Representable Functors and Universal Elements", " 6. Adjoint Functors", "Chapter 16. Multilinear Algebra", " 1. Iterated Tensor Products", " 2. Spaces of Tensors", " 3. Graded Modules", " 4. Graded Algebras", " 5. The Graded Tensor Algebra", " 6. The Exterior Algebra of a Module", " 7. Determinants by Exterior Algebras", " 8. Subspaces by Exterior Algebra", " 9. Duality in Exterior Algebra", " 10. Alternating Forms and Skew-Symmetric Tensors", "Bibliography", "Index".

En rimelig kompakt udgave af i alt fald første og andet år af et matematik-bifag. Hård kost som selvstudie.… (mer)