Differential Algebra & Algebraic Groups
Academic Press
Published on 15. June 1973
446 pages
978-0-08-087369-5 (ISBN)
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Differential Algebra & Algebraic Groups
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E.R. Kolchin
Differential Algebra and Algebraic Groups
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11/1997
Academic Press
€136.40
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Content
- Front Cover
- Differential Algebra and Algebraic Groups
- Copyright Page
- Contents
- Preface
- Acknowledgments
- Chapter 0. Algebraic Preliminaries
- 1. Conventions
- 2. Separable dependence
- 3. Quasi-separable field extensions
- 4. Quotients
- 5. Perfect ideals
- 6. Separable, quasi-separable, and regular ideals
- 7. Conservative systems
- 8. Perfect conservative systems
- 9. Noetherian conservative systems
- 10. Morphisms and birational equivalence of ideals
- 11. Polynomial ideals and generic zeros
- 12. Polynomial ideals and ground field extension
- 13. Power series
- 14. Specializations
- 15. Algebraic function fields of one variable
- 16. Dimension of components
- 17. Lattice points
- 18. Shapiro's lemma
- 19. t-Values
- Chapter I. Basic Notions of Differential Algebra
- 1. Differential rings
- 2. Homomorphisms and differential ideals
- 3. Differential rings of quotients
- 4. Transformation and restriction of the set of derivation operators
- 5. Differential modules
- differential algebras
- 6. Differential polynomial algebras
- 7. Permissible gradings
- 8. Rank
- 9. Autoreduced sets
- 10. Characteristic sets
- 11. Pseudo-leaders
- 12. Differential algebras of power series
- Chapter II. Differential Fields
- 1. Linear dependence over constants
- 2. Separable extensions
- 3. Differentially perfect and differentially quasi-perfect differential fields
- 4. Separable dependence over constants
- 5. Differential polynomial functions
- 6. Dependence of derivative operators
- 7. Differentially separable dependence
- 8. Differentially separable extensions
- 9. Differential inseparability bases
- 10. Differential transcendence bases
- 11. Finitely generated extensions
- 12. Differential inseparability polynomials
- 13. Differential type
- typical differential inseparability degree
- Chapter III. The Basis Theorem and Some Related Topics
- 1. Differential conservative systems
- 2. Quasi-separable differential ideals
- 3. Differential fields of definition
- 4. The basis theorem
- 5. Differential dimension polynomials
- 6. Extension of the differential field of coefficients
- 7. Universal extensions
- 8. t-Coherent autoreduced sets
- 9. Differential specializations
- 10. Constrained families
- Chapter IV. Algebraic Differential Equations
- PART A: CHARACTERISTIC p ARBITRARY
- 1. Differential affine space. The differential Zariski topology
- 2. Generic zeros. The theorem of zeros
- 3. Closed sets and u-separable differential ideals
- 4. The relative topologies
- differential fields of definition
- 5. Linear differential ideals
- 6. General components
- 7. General components and differential dimension polynomials
- 8. Multiplicity of zeros
- PART B: CHARACTERISTIC p = 0
- 9. Finite sets of differential polynomials
- 10. The leading coefficient theorem
- 11. Levi's lemma
- 12. The domination lemma
- 13. Preparations
- 14. The component theorem
- 15. The low power theorem
- 16. The Ritt problem
- 17. Systems of bounded order
- 18. Substitution of powers
- Bibliography for Chapters I-IV
- Chapter V. Algebraic Groups
- 1. Introduction
- 2. Pre-K-sets
- 3. K-Groups and homogeneous K-spaces. K-Sets
- 4. Extending the universal field
- 5. Extending the basic field
- 6. Zariski topology
- K-topology
- 7. Closed sets
- 8. K-Subgroups
- 9. K-Homomorphisms
- 10. Direct products
- 11. Quotients
- 12. Galois cohomology
- 13. Principal homogeneous K-spaces
- 14. Holomorphicity at a specialization
- 15. K-Mappings
- 16. K-Functions
- 17. K-Cohomology
- 18. Invariant derivations and differentials. The Lie algebra
- 19. Localrings
- 20. Tangent spaces
- 21. Crossed K-homomorphisms
- 22. Logarithmic derivatives
- 23. Linear K-groups
- 24. Abelian K-groups
- Bibliography for Chapter V
- Chapter VI. Galois Theory of Differential Fields
- 1. Specializations of isomorphisms
- 2. Strong isomorphisms
- 3. Strongly normal extensions, Galois groups
- 4. The fundamental theorems
- 5. Examples
- 6. Picard-Vessiot extensions
- 7. G-Primitives
- 8. Differential Galois cohomology
- 9. Applications
- 10. V-Primitives
- Bibliography for Chapter VI
- Glossary of Notation
- Index of Definitions
- Pure and Applied Mathematics
