Inhaltsverzeichnis
Contents vii
Preface xi
Acknowledgements xvii
Chapter 0 Algebraic Preliminaries
1 Conventions 1
2 Separable dependence 2
3 Quasi-separable field extensions 4
4 Quotients 7
5 Perfect ideals 7
6 Separable, quasi-separable, and regular ideals 8
7 Conservative systems 10
8 Perfect conservative systems 12
9 Noetherian conservative systems 13
10 Morphisms and birational equivalence of ideals 16
11 Polynomial ideals and generic zeros 19
12 Polynomial ideals and ground field extension 20
13 Power series 29
14 Specializations 33
15 Algebraic function fields of one variable 41
16 Dimension of components 43
17 Lattice points 49
18 Shapiro's lemma 53
19 k-Values 56
Chapter I Basic Notions of Differential Algebra
1 Differential rings 58
2 Homomorphisms and differential ideals 61
3 Differential rings of quotients 63
4 Transformation and restriction of the set of derivation operators 65
5 Differential modules; differential algebras 66
6 Differential polynomial algebras 69
7 Permissible gradings 72
8 Rank 75
9 Autoreduced sets 77
10 Characteristic sets 81
11 Pseudo-Ieaders 83
12 Differential algebras of power series 84
Chapter II Differential Fields
1 Linear dependence over constants 86
2 Separable extensions 89
3 Differentially perfect and differentially quasi-perfect differential fields 92
4 Separable dependence over constants 93
5 Differential polynomial functions 95
6 Dependence of derivative operators 95
7 Differentially separable dependence 99
8 Differentially separable extensions 100
9 Differential inseparability bases 104
10 Differential transcendence bases 108
11 Finitely generated extensions 109
12 Differential inseparability polynomials 115
13 Differential type; typical differential inseparability degree 118
Chapter III The Basis Theorem and Some Related Topics
1 Differential conservative systems 121
2 Quasi-separable differential ideals 123
3 Differential fields of definition 125
4 The basis theorem 126
5 Differential dimension polynomials 129
6 Extension of the differential field of coefficients 130
7 Universal extensions 133
8 k-Coherent autoreduced sets 135
9 Differential specializations 138
10 Constrained families 142
Chapter IV Algebraic Differential Equations
Part A. CHARACTERISTIC p ARBITRARY
1 Differential affine space. The differential Zariski topology 145
2 Generic zeros. The theorem of zeros 146
3 Closed sets and U-separable differential ideals 147
4 The relative topologies; differential fields of definition 148
5 Linear differential ideals 150
6 General components 155
7 General components and differential dimension polynomials 160
8 Multiplicity of zeros 164
PART B. CHARACTERISTIC p = 0
9 Finite sets of differential polynomials 166
10 The leading coefficient theorem 171
11 Levi's lemma 176
12 The domination lemma 178
13 Preparations 183
14 The component theorem 185
15 The low power theorem 187
16 The Ritt problem 190
17 Systems of bounded order 194
18 Substitution of powers 202
Bibliography for Chapters I - IV 206
Chapter V Algebraic Groups
1Introduction 212
2 Pre-K-sets 215
3 K-Groups and homogeneous K-spaces. K-Sets 218
4 Extending the universal field 227
5 Extending the basic field 230
6 Zariski topology; K-topology 236
7 Closed sets 240
8 K-Subgroups 247
9 K-Homomorphisms 249
10 Direct products 257
11 Quotients 267
12 Galois cohomology 273
13 Principal homogeneous K-spaces 281
14 Holomorphicity at a specialization 287
15 K-Mappings 294
16 K-Functions 306
17 K-Cohomology 318
18 Invariant derivations and differentials. The Lie algebra 322
19 Local rings 331
20 Tangent spaces 334
21 Crossed K-homomorphisms 341
22 Logarithmic derivatives 349
23 Linear K-groups 354
24 Abelian K-groups 376
Bibliography for Chapter V 383
Chapter VI Galois Theory of Differential Fields
1 Specializations of isomorphisms 385
2 Strong isomorphisms 388
3 Strongly normal extensions. galois groups 393
4 The fundamental theorems 398
5 Examples 404
6 Picard-Vessiot extensions 409
7 G-Primitives 418
8 Differential Galois cohomology 421
9 Applications 426
10 V-Primitives 427
Bibliography for Chapter VI 431
Glossary of Notation 435
Index of Definitions 441 - 446